OAI.ElasticityCGO.localized_formal_augmented_H1_bound: intermediate lemma from OpenAI, Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem

Proved

OAI.ElasticityCGO.localized_formal_augmented_H1_bound

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.ElasticityCGO.localized_formal_augmented_H1_bound of lean/OAI/MathematicalPhysics/Elasticity/PhysicalCGO.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.ElasticityCGO, with open Set, open scoped SchwartzMap BigOperators, open ElasticityAugmented, open ElasticityBessel (J norm2 ), open ElasticityNegativeOrder (mult sobNorm sumNorm), open ElasticityPhysicalAlgebra (divShift), open ElasticityPhysicalEstimate (physicalAug)):

theorem localized_formal_augmented_H1_bound {U : Set X} (hU : IsOpen U)
    (θ : Fin 3 → ℂ) (N : ℕ) (a : ℕ → Amplitude Smooth)
    (h₀ : normal θ (a 0).1=0)
    (hrec : ∀ j<N,EqOn (fun x => ((normal θ (a (j+1)).1 : Smooth) : X → ℂ) x)
      (fun x => ((-(div coord (a j).1-(a j).2) : Smooth) : X → ℂ) x) U)
    (χ : Smooth) (hc : HasCompactSupport (χ : X → ℂ)) (hχ : EqOn (χ : X → ℂ) 1 U)
    {g : X → ℂ} (hg : g.HasTemperateGrowth) (hs : tsupport g⊆U) :
    ∃ C : ℝ,0<C ∧ ∀ h : ℝ,0<h → h≤1 →
      sobNorm h 1 (physicalAug ((h : ℂ)⁻¹ • θ)
        (fun i => mult g (cut χ hc ((truncation (h : ℂ) N a).1 i))))≤C

Definitions. The statement is written with OAI.ElasticityAugmented.Amplitude, OAI.ElasticityAugmented.Smooth, OAI.ElasticityAugmented.X, OAI.ElasticityAugmented.coord, OAI.ElasticityAugmented.div, OAI.ElasticityAugmented.normal, OAI.ElasticityBessel.S, OAI.ElasticityBessel.X and OAI.ElasticityCGO.S (definitions bundle OAIQ6El000 of this split); OAI.ElasticityAugmented.truncation (definitions bundle OAIQ6El001 of this split); OAI.ElasticityCGO.cut and OAI.ElasticityNegativeOrder.mult (definitions bundle OAIQ6El004 of this split); OAI.ElasticityNegativeOrder.sobNorm and OAI.ElasticityPhysicalEstimate.physicalAug (definitions bundle OAIQ6El006 of this split).

Used by. The proof of OAI.ElasticityCGO.localized_actual_strong_bound of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIQ6El006

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 ElasticityCGO
open Set
open scoped SchwartzMap BigOperators
open ElasticityAugmented
open ElasticityBessel (J norm2 )
open ElasticityNegativeOrder (mult sobNorm sumNorm)
open ElasticityPhysicalAlgebra (divShift)
open ElasticityPhysicalEstimate (physicalAug)

theorem localized_formal_augmented_H1_bound {U : Set X} (hU : IsOpen U)
    (θ : Fin 3 → ℂ) (N : ℕ) (a : ℕ → Amplitude Smooth)
    (h₀ : normal θ (a 0).1=0)
    (hrec : ∀ j<N,EqOn (fun x => ((normal θ (a (j+1)).1 : Smooth) : X → ℂ) x)
      (fun x => ((-(div coord (a j).1-(a j).2) : Smooth) : X → ℂ) x) U)
    (χ : Smooth) (hc : HasCompactSupport (χ : X → ℂ)) (hχ : EqOn (χ : X → ℂ) 1 U)
    {g : X → ℂ} (hg : g.HasTemperateGrowth) (hs : tsupport g⊆U) :
    ∃ C : ℝ,0<C ∧ ∀ h : ℝ,0<h → h≤1 →
      sobNorm h 1 (physicalAug ((h : ℂ)⁻¹ • θ)
        (fun i => mult g (cut χ hc ((truncation (h : ℂ) N a).1 i))))≤C := by
  sorry
end ElasticityCGO

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/PhysicalCGO.lean (declaration OAI.ElasticityCGO.localized_formal_augmented_H1_bound)

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