OAI.Elasticity.dn_weak_transport_column: intermediate lemma from OpenAI, Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem

Proved

OAI.Elasticity.dn_weak_transport_column

by dbenbenn · Oct 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.Elasticity.global_uniqueness, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem, OpenAI Math Release, September 24, 2026, result family 372, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.Elasticity.dn_weak_transport_column of lean/OAI/MathematicalPhysics/Elasticity/WeakLimits.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.Elasticity, with open Set MeasureTheory Filter TemperedDistribution, open scoped BigOperators SchwartzMap Topology, open ElasticityAugmented, open ElasticityCoeffBounds (ExteriorConstant), open ElasticityDistribution (Coeff Dist LocalEq AugHilbert leadingRd leadingSd normald), open ElasticityPhysicalAlgebra (phase physical), open ElasticityPhysicalEstimate (physicalAug), open ElasticityNegativeOrder (sobNorm), open ElasticityCGO):

theorem dn_weak_transport_column {Ω B B' : Set X}
    (hΩ : IsOpen Ω) (hOB : Bornology.IsBounded Ω) (hB : IsOpen B)
    (hBB : Bornology.IsBounded B) (hB' : IsOpen B') (hsub : B'⊆B) (hcl : closure Ω⊆B')
    (R : ℝ) (hR : 1≤R) (hKR : closure Ω⊆Metric.closedBall 0 R)
    (hBR : B⊆Metric.ball 0 R) (α β : X) (hα : α≠0)
    (hθ : ∑ i,((α i : ℂ)+Complex.I*(β i : ℂ))*((α i : ℂ)+Complex.I*(β i : ℂ))=0)
    (lam m : Fin 2 → X → ℝ)
    (hl : ∀ q,ContDiff ℝ (⊤ : ℕ∞) (lam q)) (hm : ∀ q,ContDiff ℝ (⊤ : ℕ∞) (m q))
    (hp : ∀ q x,0 < m q x ∧ 0<3*lam q x+2*m q x)
    (hcll : ∀ q,ExteriorConstant (smoothOfReal (lam q) (hl q)))
    (hclm : ∀ q,ExteriorConstant (smoothOfReal (m q) (hm q)))
    (he : ∀ x∉Ω,lam 0 x=lam 1 x ∧ m 0 x=m 1 x)
    (hDN : DN Ω (lam 0) (m 0)=DN Ω (lam 1) (m 1))
    (χ g η : Smooth) (hcχ : HasCompactSupport (χ : X → ℂ))
    (hc : HasCompactSupport (g : X → ℂ)) (hs : tsupport (g : X → ℂ)⊆B)
    (hχ : EqOn (χ : X → ℂ) 1 B) (hg : EqOn (g : X → ℂ) 1 B')
    (hcη : HasCompactSupport (η : X → ℂ)) (hsη : tsupport (η : X → ℂ)⊆Metric.ball 0 R)
    (hη : EqOn (η : X → ℂ) 1 B)
    (a : ℕ → Amplitude Smooth)
    (ha : PhysicalColumn R hR α β (smoothOfReal (lam 0) (hl 0)) (smoothOfReal (m 0) (hm 0)) a)
    (h₀ : normal (fun i => (α i : ℂ)+Complex.I*(β i : ℂ)) (a 0).1=0)
    (hrec : ∀ j<8,EqOn (fun x => ((normal (fun i => (α i : ℂ)+Complex.I*(β i : ℂ))
      (a (j+1)).1 : Smooth) : X → ℂ) x)
      (fun x => ((-(ElasticityAugmented.div coord (a j).1-(a j).2) : Smooth) : X → ℂ) x) (Metric.ball 0 R)) :
    let L : Coeff := ⟨smoothOfReal (lam 1) (hl 1),(hcll 1).growth⟩
    let M : Coeff := ⟨smoothOfReal (m 1) (hm 1),(hclm 1).growth⟩
    let θ := fun i => (α i : ℂ)+Complex.I*(β i : ℂ)
    ∃ W : AugHilbert,
      (∀ i,LocalEq B' (leadingRd θ L M (fun j => (W j.succ : Dist)) (W 0 : Dist) i) 0) ∧
      LocalEq B' (leadingSd θ L M (fun j => (W j.succ : Dist)) (W 0 : Dist)) 0 ∧
      LocalEq B' (normald θ (fun j => (W j.succ : Dist))) 0 ∧
      ∀ i,LocalEq (closure Ω)ᶜ (W i : Dist) (localizedLeading R hR a η i : Dist)

Definitions. The statement is written with OAI.ElasticityAugmented.Amplitude, OAI.ElasticityAugmented.Smooth, OAI.ElasticityAugmented.X, OAI.ElasticityAugmented.coord, OAI.ElasticityAugmented.div, OAI.ElasticityAugmented.normal, OAI.ElasticityDistribution.Coeff, OAI.ElasticityDistribution.Dist and OAI.ElasticityDistribution.X (definitions bundle OAIQ6El000 of this split); OAI.ElasticityAugmented.smoothOfReal, OAI.ElasticityCGO.PhysicalColumn, OAI.ElasticityCGO.localizedLeading, OAI.ElasticityCoeffBounds.ExteriorConstant, OAI.ElasticityDistribution.AugHilbert, OAI.ElasticityDistribution.LocalEq and OAI.ElasticityCoeffBounds.ExteriorConstant.growth (definitions bundle OAIQ6El004 of this split); OAI.ElasticityDistribution.normald (definitions bundle OAIQ6El005 of this split); OAI.ElasticityDistribution.leadingRd and OAI.ElasticityDistribution.leadingSd (definitions bundle OAIQ6El006 of this split); OAI.Elasticity.BoundaryData, OAI.Elasticity.DN and OAI.Elasticity.X (the published bundle Def_ElasticityUniqueness).

Used by. The proofs of OAI_Elasticity_dn_common_leading_frame and OAI.Elasticity.dn_smooth_transport_column of this split import it.

Preamble

import Mathlib
import Definitions.Def_ElasticityUniqueness
import Definitions.Def_OAIQ6El006

Formal statement

section

set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

noncomputable section

namespace Elasticity
open Set MeasureTheory Filter TemperedDistribution
open scoped BigOperators SchwartzMap Topology
open ElasticityAugmented
open ElasticityCoeffBounds (ExteriorConstant)
open ElasticityDistribution (Coeff Dist LocalEq AugHilbert leadingRd leadingSd normald)
open ElasticityPhysicalAlgebra (phase physical)
open ElasticityPhysicalEstimate (physicalAug)
open ElasticityNegativeOrder (sobNorm)
open ElasticityCGO

theorem dn_weak_transport_column {Ω B B' : Set X}
    (hΩ : IsOpen Ω) (hOB : Bornology.IsBounded Ω) (hB : IsOpen B)
    (hBB : Bornology.IsBounded B) (hB' : IsOpen B') (hsub : B'⊆B) (hcl : closure Ω⊆B')
    (R : ℝ) (hR : 1≤R) (hKR : closure Ω⊆Metric.closedBall 0 R)
    (hBR : B⊆Metric.ball 0 R) (α β : X) (hα : α≠0)
    (hθ : ∑ i,((α i : ℂ)+Complex.I*(β i : ℂ))*((α i : ℂ)+Complex.I*(β i : ℂ))=0)
    (lam m : Fin 2 → X → ℝ)
    (hl : ∀ q,ContDiff ℝ (⊤ : ℕ∞) (lam q)) (hm : ∀ q,ContDiff ℝ (⊤ : ℕ∞) (m q))
    (hp : ∀ q x,0 < m q x ∧ 0<3*lam q x+2*m q x)
    (hcll : ∀ q,ExteriorConstant (smoothOfReal (lam q) (hl q)))
    (hclm : ∀ q,ExteriorConstant (smoothOfReal (m q) (hm q)))
    (he : ∀ x∉Ω,lam 0 x=lam 1 x ∧ m 0 x=m 1 x)
    (hDN : DN Ω (lam 0) (m 0)=DN Ω (lam 1) (m 1))
    (χ g η : Smooth) (hcχ : HasCompactSupport (χ : X → ℂ))
    (hc : HasCompactSupport (g : X → ℂ)) (hs : tsupport (g : X → ℂ)⊆B)
    (hχ : EqOn (χ : X → ℂ) 1 B) (hg : EqOn (g : X → ℂ) 1 B')
    (hcη : HasCompactSupport (η : X → ℂ)) (hsη : tsupport (η : X → ℂ)⊆Metric.ball 0 R)
    (hη : EqOn (η : X → ℂ) 1 B)
    (a : ℕ → Amplitude Smooth)
    (ha : PhysicalColumn R hR α β (smoothOfReal (lam 0) (hl 0)) (smoothOfReal (m 0) (hm 0)) a)
    (h₀ : normal (fun i => (α i : ℂ)+Complex.I*(β i : ℂ)) (a 0).1=0)
    (hrec : ∀ j<8,EqOn (fun x => ((normal (fun i => (α i : ℂ)+Complex.I*(β i : ℂ))
      (a (j+1)).1 : Smooth) : X → ℂ) x)
      (fun x => ((-(ElasticityAugmented.div coord (a j).1-(a j).2) : Smooth) : X → ℂ) x) (Metric.ball 0 R)) :
    let L : Coeff := ⟨smoothOfReal (lam 1) (hl 1),(hcll 1).growth⟩
    let M : Coeff := ⟨smoothOfReal (m 1) (hm 1),(hclm 1).growth⟩
    let θ := fun i => (α i : ℂ)+Complex.I*(β i : ℂ)
    ∃ W : AugHilbert,
      (∀ i,LocalEq B' (leadingRd θ L M (fun j => (W j.succ : Dist)) (W 0 : Dist) i) 0) ∧
      LocalEq B' (leadingSd θ L M (fun j => (W j.succ : Dist)) (W 0 : Dist)) 0 ∧
      LocalEq B' (normald θ (fun j => (W j.succ : Dist))) 0 ∧
      ∀ i,LocalEq (closure Ω)ᶜ (W i : Dist) (localizedLeading R hR a η i : Dist) := by
  sorry
end Elasticity

end

end OAI
end

Source

OpenAI, Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem, OpenAI Math Release, September 24, 2026, https://github.com/openai/math (result family 372; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/article.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/MathematicalPhysics/Elasticity/WeakLimits.lean (declaration OAI.Elasticity.dn_weak_transport_column)

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