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

Proved

OAI.ElasticityDistribution.normalized_quantitative_gain

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.ElasticityDistribution.normalized_quantitative_gain of lean/OAI/MathematicalPhysics/Elasticity/CGO.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

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

theorem normalized_quantitative_gain (k : ℤ) {Ω : Set X} {χ η ξ : X → ℂ}
    (hχ : ContDiff ℝ (⊤ : ℕ∞) χ) (hcχ : HasCompactSupport χ)
    (hη : ContDiff ℝ (⊤ : ℕ∞) η) (hcη : HasCompactSupport η)
    (hξ : ContDiff ℝ (⊤ : ℕ∞) ξ) (hcξ : HasCompactSupport ξ)
    (h1 : Set.EqOn η 1 (tsupport χ)) (h2 : Set.EqOn ξ 1 (tsupport η))
    (hsη : tsupport η ⊆ Ω) (lam m n o : Coeff) (hn : n*m=1) (ho : o*(lam+m+m)=1) :
    ∃ C : ℝ, 0<C ∧ ∀ (z : Fin 3 → ℂ) (U F : Fin 4 → Dist) (f g : HV),
      (∀ i, LocalEq Ω (normPair z lam m n o (fun j => U j.succ) (U 0) i) (F i)) →
      (∀ i, sobolevInjection (k : ℝ) (f i)=smulLeftCLM ℂ ξ (U i)) →
      (∀ i, sobolevInjection ((k : ℝ)-1) (g i)=smulLeftCLM ℂ η (F i)) →
      ∃ q : HV, (∀ i, sobolevInjection ((k : ℝ)+1) (q i)=smulLeftCLM ℂ χ (U i)) ∧
        ‖q‖ ≤ C*(1+∑ d, ‖z d‖)^2*(‖f‖+‖g‖)

Definitions. The statement is written with OAI.ElasticityAugmented.Smooth, OAI.ElasticityAugmented.X, OAI.ElasticityDistribution.Coeff, OAI.ElasticityDistribution.Dist and OAI.ElasticityDistribution.X (definitions bundle OAIQ6El000 of this split); OAI.ElasticityRegularity.sobolevInjection (definitions bundle OAIQ6El002 of this split); OAI.ElasticityDistribution.H, OAI.ElasticityDistribution.HV, OAI.ElasticityDistribution.LocalEq and OAI.ElasticityDistribution.normPair (definitions bundle OAIQ6El004 of this split).

Used by. The proof of OAI.ElasticityDistribution.physical_quantitative_H1 of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIQ6El004

Formal statement

section

attribute [-instance] _root_.OAI.ElasticityGeneralSlicing.cylinder_temperate
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 ElasticityDistribution
open MeasureTheory TemperedDistribution

section
open scoped SchwartzMap LineDeriv Laplacian BigOperators
open ElasticityAugmented ElasticityRegularity

theorem normalized_quantitative_gain (k : ℤ) {Ω : Set X} {χ η ξ : X → ℂ}
    (hχ : ContDiff ℝ (⊤ : ℕ∞) χ) (hcχ : HasCompactSupport χ)
    (hη : ContDiff ℝ (⊤ : ℕ∞) η) (hcη : HasCompactSupport η)
    (hξ : ContDiff ℝ (⊤ : ℕ∞) ξ) (hcξ : HasCompactSupport ξ)
    (h1 : Set.EqOn η 1 (tsupport χ)) (h2 : Set.EqOn ξ 1 (tsupport η))
    (hsη : tsupport η ⊆ Ω) (lam m n o : Coeff) (hn : n*m=1) (ho : o*(lam+m+m)=1) :
    ∃ C : ℝ, 0<C ∧ ∀ (z : Fin 3 → ℂ) (U F : Fin 4 → Dist) (f g : HV),
      (∀ i, LocalEq Ω (normPair z lam m n o (fun j => U j.succ) (U 0) i) (F i)) →
      (∀ i, sobolevInjection (k : ℝ) (f i)=smulLeftCLM ℂ ξ (U i)) →
      (∀ i, sobolevInjection ((k : ℝ)-1) (g i)=smulLeftCLM ℂ η (F i)) →
      ∃ q : HV, (∀ i, sobolevInjection ((k : ℝ)+1) (q i)=smulLeftCLM ℂ χ (U i)) ∧
        ‖q‖ ≤ C*(1+∑ d, ‖z d‖)^2*(‖f‖+‖g‖) := by
  sorry
end
open scoped SchwartzMap LineDeriv Laplacian BigOperators ENNReal
open ElasticityAugmented ElasticityRegularity
end ElasticityDistribution

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/CGO.lean (declaration OAI.ElasticityDistribution.normalized_quantitative_gain)

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