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

Proved

OAI.ElasticityDistribution.smooth_column_patch

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.ElasticityDistribution.smooth_column_patch of lean/OAI/MathematicalPhysics/Elasticity/Smoothing.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.ElasticityDistribution, with open Set Filter TemperedDistribution, open scoped SchwartzMap LineDeriv BigOperators Topology, open ElasticityAugmented, open ElasticityPhysicalAlgebra (toSmooth)):

theorem smooth_column_patch {K B : Set X} (hB : IsOpen B) (hKB : K⊆B)
    (θ : Fin 3 → ℂ) (lam m : Coeff) (v φ : Fin 4 → Test) (a₀ : Fin 4 → Smooth)
    (hs : ∀ i,tsupport (v i-φ i : Test)⊆K)
    (hφ : ∀ i,EqOn (φ i : X → ℂ) (a₀ i : X → ℂ) B)
    (hr : ∀ i,LocalEq B (leadingRd θ lam m (fun j => (v j.succ : Dist)) (v 0 : Dist) i) 0)
    (he : LocalEq B (leadingSd θ lam m (fun j => (v j.succ : Dist)) (v 0 : Dist)) 0)
    (hn : LocalEq B (normald θ (fun j => (v j.succ : Dist))) 0)
    (hn₀ : normal θ (fun j => a₀ j.succ)=0) :
    ∃ a : Fin 4 → Smooth,
      (∀ i,tsupport ((a i-a₀ i : Smooth) : X → ℂ)⊆K) ∧
      (∀ i,EqOn (leadingR (A := Smooth) coord θ lam.val m.val (fun j => a j.succ) (a 0) i : X → ℂ) 0 B) ∧
      EqOn (leadingS (A := Smooth) coord θ lam.val m.val (fun j => a j.succ) (a 0) : X → ℂ) 0 B ∧
      normal θ (fun j => a j.succ)=0

Definitions. The statement is written with OAI.ElasticityAugmented.Smooth, OAI.ElasticityAugmented.X, OAI.ElasticityAugmented.coord, OAI.ElasticityAugmented.leadingR, OAI.ElasticityAugmented.leadingS, OAI.ElasticityAugmented.normal, OAI.ElasticityDistribution.Coeff, OAI.ElasticityDistribution.Dist and OAI.ElasticityDistribution.X (definitions bundle OAIQ6El000 of this split); OAI.ElasticityDistribution.LocalEq and OAI.ElasticityDistribution.Test (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).

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_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

open Set Filter TemperedDistribution
open scoped SchwartzMap LineDeriv BigOperators Topology
namespace ElasticityDistribution
open ElasticityAugmented
open ElasticityPhysicalAlgebra (toSmooth)
theorem smooth_column_patch {K B : Set X} (hB : IsOpen B) (hKB : K⊆B)
    (θ : Fin 3 → ℂ) (lam m : Coeff) (v φ : Fin 4 → Test) (a₀ : Fin 4 → Smooth)
    (hs : ∀ i,tsupport (v i-φ i : Test)⊆K)
    (hφ : ∀ i,EqOn (φ i : X → ℂ) (a₀ i : X → ℂ) B)
    (hr : ∀ i,LocalEq B (leadingRd θ lam m (fun j => (v j.succ : Dist)) (v 0 : Dist) i) 0)
    (he : LocalEq B (leadingSd θ lam m (fun j => (v j.succ : Dist)) (v 0 : Dist)) 0)
    (hn : LocalEq B (normald θ (fun j => (v j.succ : Dist))) 0)
    (hn₀ : normal θ (fun j => a₀ j.succ)=0) :
    ∃ a : Fin 4 → Smooth,
      (∀ i,tsupport ((a i-a₀ i : Smooth) : X → ℂ)⊆K) ∧
      (∀ i,EqOn (leadingR (A := Smooth) coord θ lam.val m.val (fun j => a j.succ) (a 0) i : X → ℂ) 0 B) ∧
      EqOn (leadingS (A := Smooth) coord θ lam.val m.val (fun j => a j.succ) (a 0) : X → ℂ) 0 B ∧
      normal θ (fun j => a j.succ)=0 := by
  sorry
end ElasticityDistribution

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/Smoothing.lean (declaration OAI.ElasticityDistribution.smooth_column_patch)

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