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

Proved

OAI.ElasticityPlanar.exists_spatial_transport

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.ElasticityPlanar.exists_spatial_transport of lean/OAI/MathematicalPhysics/Elasticity/Transport.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.ElasticityPlanar, with open Set Complex Module, open scoped BigOperators):

theorem exists_spatial_transport (J : ℂ →L[ℝ] E) (hJ : Function.Injective J)
    (a : n → n → E → ℂ) (ha : ∀ i j, ContDiff ℝ (⊤ : ℕ∞) (a i j))
    (f : n → E → ℂ) (hf : ∀ i, ContDiff ℝ (⊤ : ℕ∞) (f i))
    {C : Set E} (hC : IsCompact C) :
    ∃ u : n → E → ℂ, (∀ i, ContDiff ℝ (⊤ : ℕ∞) (u i)) ∧
      ∀ i x, x∈C → fderiv ℝ (u i) x (J 1)+I*fderiv ℝ (u i) x (J I)+
        ∑ j, a i j x*u j x=f i x

Used by. The proof of OAI.ElasticityFrame.exists_constraint_transport of this split imports it.

Preamble

import Mathlib

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 ElasticityPlanar
open Set Complex Module
open scoped BigOperators
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]

section
variable {n : Type*} [Fintype n]

theorem exists_spatial_transport (J : ℂ →L[ℝ] E) (hJ : Function.Injective J)
    (a : n → n → E → ℂ) (ha : ∀ i j, ContDiff ℝ (⊤ : ℕ∞) (a i j))
    (f : n → E → ℂ) (hf : ∀ i, ContDiff ℝ (⊤ : ℕ∞) (f i))
    {C : Set E} (hC : IsCompact C) :
    ∃ u : n → E → ℂ, (∀ i, ContDiff ℝ (⊤ : ℕ∞) (u i)) ∧
      ∀ i x, x∈C → fderiv ℝ (u i) x (J 1)+I*fderiv ℝ (u i) x (J I)+
        ∑ j, a i j x*u j x=f i x := by
  sorry
end
end ElasticityPlanar

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/Transport.lean (declaration OAI.ElasticityPlanar.exists_spatial_transport)

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