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

Proved

OAI.Elasticity.physical_classical_transfer

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.Elasticity.physical_classical_transfer of lean/OAI/MathematicalPhysics/Elasticity/WeakSolutions.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.Elasticity, with open MeasureTheory Set SchwartzMap TemperedDistribution, open scoped BigOperators SchwartzMap LineDeriv, open ElasticityDistribution):

theorem physical_classical_transfer
    (ha₁ : Admissible Ω lam₁ mu₁) (ha₂ : Admissible Ω lam₂ mu₂)
    (hΩ : IsOpen Ω) (hOB : Bornology.IsBounded Ω) (hΩB : Ω⊆B)
    (hB : IsOpen B) (hB' : IsOpen B') (hsub : B'⊆B)
    (hl₁ : BoundedCoefficient B lam₁) (hm₁ : BoundedCoefficient B mu₁)
    (hl₂ : BoundedCoefficient B lam₂) (hm₂ : BoundedCoefficient B mu₂)
    (he : ∀ x∉Ω, lam₁ x=lam₂ x ∧ mu₁ x=mu₂ x)
    (hDN : DN Ω lam₁ mu₁=DN Ω lam₂ mu₂)
    (L₁ M₁ L₂ M₂ N₂ O₂ : Coeff) (hN : N₂*M₂=1) (hO : O₂*(L₂+M₂+M₂)=1)
    (hL₁ : ∀ᵐ x ∂(volume.restrict B), eval L₁ x=(lam₁ x : ℂ))
    (hM₁ : ∀ᵐ x ∂(volume.restrict B), eval M₁ x=(mu₁ x : ℂ))
    (hL₂ : ∀ᵐ x ∂(volume.restrict B), eval L₂ x=(lam₂ x : ℂ))
    (hM₂ : ∀ᵐ x ∂(volume.restrict B), eval M₂ x=(mu₂ x : ℂ))
    {g : X → ℂ} (hg : ContDiff ℝ (⊤ : ℕ∞) g) (hc : HasCompactSupport g)
    (hs : tsupport g⊆B) (h1 : Set.EqOn g 1 B') (F : Fin 3 → SchwartzMap X ℂ)
    (hF : ∀ i, Set.EqOn (ElasticityPhysicalAlgebra.physical 0 (eval L₁) (eval M₁) F i : X → ℂ) 0 B) :
    ∃ G : Fin 3 → SchwartzMap X ℂ,
      (∀ i, Set.EqOn (ElasticityPhysicalAlgebra.physical 0 (eval L₂) (eval M₂) G i : X → ℂ) 0 B') ∧
      ∀ i, Set.EqOn (G i : X → ℂ) (F i) (B'\closure Ω)

Definitions. The statement is written with OAI.Elasticity.BoundedCoefficient, OAI.ElasticityAugmented.Smooth, OAI.ElasticityAugmented.X, OAI.ElasticityBessel.S, OAI.ElasticityBessel.X, OAI.ElasticityDistribution.Coeff, OAI.ElasticityDistribution.X and OAI.ElasticityDistribution.eval (definitions bundle OAIQ6El000 of this split); OAI.ElasticityPhysicalAlgebra.physical (definitions bundle OAIQ6El004 of this split); OAI.Elasticity.Admissible, OAI.Elasticity.BoundaryData, OAI.Elasticity.DN and OAI.Elasticity.X (the published bundle Def_ElasticityUniqueness).

Used by. The proofs of OAI.Elasticity.extended_dn_amplitude_transfer and OAI_Elasticity_physical_supported_transfer of this split import it.

Preamble

import Mathlib
import Definitions.Def_ElasticityUniqueness
import Definitions.Def_OAIQ6El004

Formal statement

section

attribute [-instance] _root_.OAI.ElasticityGeneralSlicing.cylinder_temperate _root_.OAI.ElasticityODE.instFactLeRealOfNat_solutions
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 MeasureTheory Set SchwartzMap TemperedDistribution
open scoped BigOperators SchwartzMap LineDeriv
namespace Elasticity
open ElasticityDistribution
variable {Ω B B' : Set X} {lam₁ mu₁ lam₂ mu₂ : X → ℝ}
theorem physical_classical_transfer
    (ha₁ : Admissible Ω lam₁ mu₁) (ha₂ : Admissible Ω lam₂ mu₂)
    (hΩ : IsOpen Ω) (hOB : Bornology.IsBounded Ω) (hΩB : Ω⊆B)
    (hB : IsOpen B) (hB' : IsOpen B') (hsub : B'⊆B)
    (hl₁ : BoundedCoefficient B lam₁) (hm₁ : BoundedCoefficient B mu₁)
    (hl₂ : BoundedCoefficient B lam₂) (hm₂ : BoundedCoefficient B mu₂)
    (he : ∀ x∉Ω, lam₁ x=lam₂ x ∧ mu₁ x=mu₂ x)
    (hDN : DN Ω lam₁ mu₁=DN Ω lam₂ mu₂)
    (L₁ M₁ L₂ M₂ N₂ O₂ : Coeff) (hN : N₂*M₂=1) (hO : O₂*(L₂+M₂+M₂)=1)
    (hL₁ : ∀ᵐ x ∂(volume.restrict B), eval L₁ x=(lam₁ x : ℂ))
    (hM₁ : ∀ᵐ x ∂(volume.restrict B), eval M₁ x=(mu₁ x : ℂ))
    (hL₂ : ∀ᵐ x ∂(volume.restrict B), eval L₂ x=(lam₂ x : ℂ))
    (hM₂ : ∀ᵐ x ∂(volume.restrict B), eval M₂ x=(mu₂ x : ℂ))
    {g : X → ℂ} (hg : ContDiff ℝ (⊤ : ℕ∞) g) (hc : HasCompactSupport g)
    (hs : tsupport g⊆B) (h1 : Set.EqOn g 1 B') (F : Fin 3 → SchwartzMap X ℂ)
    (hF : ∀ i, Set.EqOn (ElasticityPhysicalAlgebra.physical 0 (eval L₁) (eval M₁) F i : X → ℂ) 0 B) :
    ∃ G : Fin 3 → SchwartzMap X ℂ,
      (∀ i, Set.EqOn (ElasticityPhysicalAlgebra.physical 0 (eval L₂) (eval M₂) G i : X → ℂ) 0 B') ∧
      ∀ i, Set.EqOn (G i : X → ℂ) (F i) (B'\closure Ω) := by
  sorry
end Elasticity

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/WeakSolutions.lean (declaration OAI.Elasticity.physical_classical_transfer)

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