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

Proved

OAI.ElasticityBoundaryCutoffParametrix.first_scaled_identity

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.ElasticityBoundaryCutoffParametrix.first_scaled_identity of lean/OAI/MathematicalPhysics/Elasticity/BoundaryParametrix.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.ElasticityBoundaryCutoffParametrix, with open Set MeasureTheory Filter, open scoped BigOperators Topology, open MvPolynomial ElasticityBoundaryMVCalculus ElasticityBoundaryMVNormal):

lemma first_scaled_identity {χ : Y → ℂ} (hχ : ContDiff ℝ (⊤ : ℕ∞) χ)
    {δ : ℝ} (hδ : δ≠0) (q : Polynomial W) (j β : Fin 3) (y : Y) :
    dpart (cutoffLayer χ q δ j) β (scale δ y)=
      ((δ^2)⁻¹:ℝ) • (envelope δ y*firstScaled χ q δ j β y)

Definitions. The statement is written with OAI.ElasticityBoundaryCutoffLayer.cutoffLayer, OAI.ElasticityBoundaryCutoffParametrix.Y, OAI.ElasticityBoundaryCutoffParametrix.firstScaled, OAI.ElasticityBoundaryMVCalculus.MP, OAI.ElasticityBoundaryMVNormal.W, OAI.ElasticityBoundaryProfile.envelope, OAI.ElasticityBoundaryScaling.Y, OAI.ElasticityBoundaryScaling.scale and OAI.ElasticityBoundaryStrongLayer.dpart (definitions bundle OAIQ6El001 of this split).

Used by. The proofs of OAI.ElasticityBoundaryMixedScaling.mixedEnergy_limit and OAI.ElasticityBoundaryPacket.Setup.field_gradient_mass of this split import it.

Preamble

import Mathlib
import Definitions.Def_OAIQ6El001

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 MeasureTheory Filter
open scoped BigOperators Topology
namespace ElasticityBoundaryCutoffParametrix
open MvPolynomial ElasticityBoundaryMVCalculus ElasticityBoundaryMVNormal
  ElasticityBoundaryParameter ElasticityBoundaryParameterBounds ElasticityBoundaryTensorSeries
  ElasticityBoundaryActualTower ElasticityBoundarySmoothOperator ElasticityBoundaryProfile
  ElasticityBoundaryProfileBounds ElasticityBoundaryScaling ElasticityBoundaryLayerOperator
  ElasticityBoundaryWeights ElasticityBoundaryStrongLayer ElasticityBoundaryOrderBound
  ElasticityBoundaryCutoffBound ElasticityBoundaryCutoffLayer

lemma first_scaled_identity {χ : Y → ℂ} (hχ : ContDiff ℝ (⊤ : ℕ∞) χ)
    {δ : ℝ} (hδ : δ≠0) (q : Polynomial W) (j β : Fin 3) (y : Y) :
    dpart (cutoffLayer χ q δ j) β (scale δ y)=
      ((δ^2)⁻¹:ℝ) • (envelope δ y*firstScaled χ q δ j β y) := by
  sorry

end ElasticityBoundaryCutoffParametrix

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/BoundaryParametrix.lean (declaration OAI.ElasticityBoundaryCutoffParametrix.first_scaled_identity)

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