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

Proved

OAI.ElasticityBoundaryMVTower.triangular_amplitudes

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.ElasticityBoundaryMVTower.triangular_amplitudes of lean/OAI/MathematicalPhysics/Elasticity/BoundarySymbols.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.ElasticityBoundaryMVTower, with open scoped BigOperators, open ElasticityBoundaryMVNormal):

theorem triangular_amplitudes (l m : ℂ) (hm : m≠0) (he : l+2*m≠0) (hp : l+3*m≠0)
    (A : ℕ → (W →ₗ[ℂ] W)) (hA : A 0=normal l m)
    (p₀ : W) (h₀ : normal l m p₀=0) :
    ∃ p : ℕ → W, p 0=p₀ ∧
      (∀ n, 0<n → boundary (p n)=0) ∧
      ∀ n, ∑ k ∈ Finset.range (n+1), A k (p (n-k))=0

Definitions. The statement is written with OAI.ElasticityBoundaryMVCalculus.MP, OAI.ElasticityBoundaryMVNormal.W, OAI.ElasticityBoundaryMVNormal.boundary and OAI.ElasticityBoundaryMVNormal.normal (definitions bundle OAIQ6El001 of this split).

Used by. The proof of OAI.ElasticityBoundaryActualTower.finite_tensor_amplitude of this split imports 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 BigOperators
namespace ElasticityBoundaryMVTower
open ElasticityBoundaryMVNormal

theorem triangular_amplitudes (l m : ℂ) (hm : m≠0) (he : l+2*m≠0) (hp : l+3*m≠0)
    (A : ℕ → (W →ₗ[ℂ] W)) (hA : A 0=normal l m)
    (p₀ : W) (h₀ : normal l m p₀=0) :
    ∃ p : ℕ → W, p 0=p₀ ∧
      (∀ n, 0<n → boundary (p n)=0) ∧
      ∀ n, ∑ k ∈ Finset.range (n+1), A k (p (n-k))=0 := by
  sorry

end ElasticityBoundaryMVTower

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/BoundarySymbols.lean (declaration OAI.ElasticityBoundaryMVTower.triangular_amplitudes)

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