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

Proved

OAI.ElasticityPlanar.exists_algebra_variable_homotopy

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_algebra_variable_homotopy of lean/OAI/MathematicalPhysics/Elasticity/Homotopy.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 SchwartzMap LineDeriv BigOperators, open ElasticityRegularity):

theorem exists_algebra_variable_homotopy {S Ω Ω' K : Set ℂ} (hS : IsCompact S)
    (hΩS : Ω ⊆ S) (hΩ' : IsOpen Ω') (hsub : Ω' ⊆ Ω)
    (hK : IsCompact K) (q : Basis n ℂ R) (A : P × ℂ → R) (hA : ContDiff ℝ (⊤ : ℕ∞) A)
    (hAc : ∀ p z, z∉K → A (p,z)=0) {g : ℂ → ℂ} (hg : ContDiff ℝ (⊤ : ℕ∞) g)
    (hc : HasCompactSupport g) (hs : tsupport g ⊆ Ω) (h1 : Set.EqOn g 1 Ω') :
    ∃ B : (ℝ × P) × ℂ → R, ContDiff ℝ (⊤ : ℕ∞) B ∧
      ∀ s p z, z∈Ω' → classicalDbar (fun z => B ((s,p),z)) z=
        A (p,z)+s • (A (p,z)*B ((s,p),z)-B ((s,p),z)*A (p,z))

Definitions. The statement is written with OAI.ElasticityPlanar.classicalDbar (definitions bundle OAIQ6El003 of this split).

Used by. The proof of OAI.ElasticityPlanar.spatial_frame_of_left_inverse of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIQ6El003

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 Set Complex Module
open scoped SchwartzMap LineDeriv BigOperators
namespace ElasticityPlanar
open ElasticityRegularity
variable {R : Type*} [NormedRing R] [NormedAlgebra ℂ R] [CompleteSpace R]
  [FiniteDimensional ℂ R]
variable {n : Type*} [Fintype n] [DecidableEq n]
variable {P : Type*} [NormedAddCommGroup P] [NormedSpace ℝ P] [FiniteDimensional ℝ P]
omit [CompleteSpace R] [DecidableEq n] in
theorem exists_algebra_variable_homotopy {S Ω Ω' K : Set ℂ} (hS : IsCompact S)
    (hΩS : Ω ⊆ S) (hΩ' : IsOpen Ω') (hsub : Ω' ⊆ Ω)
    (hK : IsCompact K) (q : Basis n ℂ R) (A : P × ℂ → R) (hA : ContDiff ℝ (⊤ : ℕ∞) A)
    (hAc : ∀ p z, z∉K → A (p,z)=0) {g : ℂ → ℂ} (hg : ContDiff ℝ (⊤ : ℕ∞) g)
    (hc : HasCompactSupport g) (hs : tsupport g ⊆ Ω) (h1 : Set.EqOn g 1 Ω') :
    ∃ B : (ℝ × P) × ℂ → R, ContDiff ℝ (⊤ : ℕ∞) B ∧
      ∀ s p z, z∈Ω' → classicalDbar (fun z => B ((s,p),z)) z=
        A (p,z)+s • (A (p,z)*B ((s,p),z)-B ((s,p),z)*A (p,z)) := by
  sorry
end ElasticityPlanar

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/Homotopy.lean (declaration OAI.ElasticityPlanar.exists_algebra_variable_homotopy)

View graph