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

Proved

OAI.ElasticityBoundaryPacket.pair_separates

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.ElasticityBoundaryPacket.pair_separates of lean/OAI/MathematicalPhysics/Elasticity/BoundaryValues.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.ElasticityBoundaryPacket, with open scoped Matrix BigOperators, open ElasticityBoundaryLeading ElasticityBoundaryWeights):

lemma pair_separates (n : ℕ) (l₁ m₁ l₂ m₂ D L M : ℂ)
    (hm₁ : m₁≠0) (hm₂ : m₂≠0) (hp₁ : l₁+3*m₁≠0) (hp₂ : l₂+3*m₂≠0)
    (hD : D≠0)
    (h : ∀ p q : W, normal l₁ m₁ p=0 → normal l₂ m₂ q=0 →
      frozenPair n (fun i α j β => D*elasticityTensor L M i α j β) p q=0) : L=0 ∧ M=0

Definitions. The statement is written with OAI.ElasticityBoundaryLayerOperator.elasticityTensor, OAI.ElasticityBoundaryMVCalculus.MP, OAI.ElasticityBoundaryMVNormal.W, OAI.ElasticityBoundaryMVNormal.normal, OAI.ElasticityBoundaryPacket.I and OAI.ElasticityBoundaryPacket.frozenPair (definitions bundle OAIQ6El001 of this split).

Used by. The proofs of OAI.ElasticityBoundaryPacket.Setup.boundary_normal_step and OAI.ElasticityBoundaryPacket.boundary_values_global of this split import it.

Preamble

import Mathlib
import Definitions.Def_OAIQ6El001

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

section

noncomputable section
open scoped Matrix BigOperators
namespace ElasticityBoundaryPacket
open ElasticityBoundaryLeading ElasticityBoundaryWeights
  ElasticityBoundarySymbol ElasticityBoundaryMVNormal ElasticityBoundaryParameter ElasticityBoundaryLayerOperator

lemma pair_separates (n : ℕ) (l₁ m₁ l₂ m₂ D L M : ℂ)
    (hm₁ : m₁≠0) (hm₂ : m₂≠0) (hp₁ : l₁+3*m₁≠0) (hp₂ : l₂+3*m₂≠0)
    (hD : D≠0)
    (h : ∀ p q : W, normal l₁ m₁ p=0 → normal l₂ m₂ q=0 →
      frozenPair n (fun i α j β => D*elasticityTensor L M i α j β) p q=0) : L=0 ∧ M=0 := by
  sorry

end ElasticityBoundaryPacket

end
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/BoundaryValues.lean (declaration OAI.ElasticityBoundaryPacket.pair_separates)

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