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

Proved

OAI.ElasticityDistribution.leading_matrix_local

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.leading_matrix_local 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 TemperedDistribution, open scoped SchwartzMap LineDeriv BigOperators, open ElasticityAugmented):

lemma leading_matrix_local {B : Set X} (θ : Fin 3 → ℂ) (hθ : ∑ i,θ i*θ i=0)
    (lam m n o : Coeff) (hn : n*m=1) (ho : o*(lam+m+m)=1) (U : Fin 4 → Dist)
    (hr : ∀ i,LocalEq B (leadingRd θ lam m (fun j => U j.succ) (U 0) i) 0)
    (hs : LocalEq B (leadingSd θ lam m (fun j => U j.succ) (U 0)) 0) (i : Fin 4) :
    LocalEq B ((2 : ℂ) • directiond θ (U i)+
      ∑ k,∑ j,θ k • (coeffA dc lam m n o k i j • U j)) 0

Definitions. The statement is written with OAI.ElasticityAugmented.Smooth, OAI.ElasticityAugmented.X, OAI.ElasticityAugmented.coeffA, OAI.ElasticityDistribution.Coeff, OAI.ElasticityDistribution.Dist and OAI.ElasticityDistribution.X (definitions bundle OAIQ6El000 of this split); OAI.ElasticityDistribution.LocalEq, OAI.ElasticityDistribution.dc, OAI.ElasticityDistribution.directiond and OAI.ElasticityDistribution.instSMulSubtypeForallXComplexMemSubalgebraSmoothCoeffDist (definitions bundle OAIQ6El004 of this split); OAI.ElasticityDistribution.leadingRd and OAI.ElasticityDistribution.leadingSd (definitions bundle OAIQ6El006 of this split).

Used by. The proofs of OAI_ElasticityDistribution_weak_leading_smooth 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

namespace ElasticityDistribution
open Set TemperedDistribution
open scoped SchwartzMap LineDeriv BigOperators
open ElasticityAugmented

lemma leading_matrix_local {B : Set X} (θ : Fin 3 → ℂ) (hθ : ∑ i,θ i*θ i=0)
    (lam m n o : Coeff) (hn : n*m=1) (ho : o*(lam+m+m)=1) (U : Fin 4 → Dist)
    (hr : ∀ i,LocalEq B (leadingRd θ lam m (fun j => U j.succ) (U 0) i) 0)
    (hs : LocalEq B (leadingSd θ lam m (fun j => U j.succ) (U 0)) 0) (i : Fin 4) :
    LocalEq B ((2 : ℂ) • directiond θ (U i)+
      ∑ k,∑ j,θ k • (coeffA dc lam m n o k i j • U j)) 0 := by
  sorry
end ElasticityDistribution
open Set Filter TemperedDistribution
open scoped SchwartzMap LineDeriv BigOperators Topology

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.leading_matrix_local)

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