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

Proved

OAI.ElasticityBoundaryPacketMass.cutoff_value_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.ElasticityBoundaryPacketMass.cutoff_value_bound of lean/OAI/MathematicalPhysics/Elasticity/BoundaryPackets.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.ElasticityBoundaryPacketMass, with open Set MeasureTheory, open scoped BigOperators, open ElasticityBoundaryCutoffParametrix ElasticityBoundaryCutoffLayer):

lemma cutoff_value_bound {χ : Y → ℂ} (hχ : ContDiff ℝ (⊤ : ℕ∞) χ)
    (hc : HasCompactSupport χ) (q : Polynomial W) (j : I) :
    ∃ C : ℝ, 0≤C ∧ ∀ δ : ℝ, 0<δ → δ≤1 →
      MemLp (cutoffLayer χ q δ j) 2 (volume.restrict halfspace) ∧
      (∫ z in halfspace, ‖cutoffLayer χ q δ j z‖^2)≤C*δ^4

Definitions. The statement is written with OAI.ElasticityBoundaryCutoffLayer.Y, OAI.ElasticityBoundaryCutoffLayer.cutoffLayer, OAI.ElasticityBoundaryMVCalculus.MP, OAI.ElasticityBoundaryMVNormal.W, OAI.ElasticityBoundaryPacketMass.I, OAI.ElasticityBoundaryPacketMass.Y, OAI.ElasticityBoundaryProfileBounds.Y and OAI.ElasticityBoundaryProfileBounds.halfspace (definitions bundle OAIQ6El001 of this split).

Used by. The proof of OAI.ElasticityBoundaryPacket.Setup.trial_size of this split imports 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
open scoped BigOperators
namespace ElasticityBoundaryPacketMass
open ElasticityBoundaryCutoffParametrix ElasticityBoundaryCutoffLayer
  ElasticityBoundaryProfile ElasticityBoundaryParameter ElasticityBoundaryParameterBounds
  ElasticityBoundaryOrderBound ElasticityBoundaryCutoffBound ElasticityBoundaryScaling
  ElasticityBoundaryWeights ElasticityBoundaryStrongLayer ElasticityBoundaryMVCalculus
  ElasticityBoundaryTensorSeries ElasticityBoundaryProfileBounds ElasticityBoundaryMVNormal ElasticityBoundaryActualTower

lemma cutoff_value_bound {χ : Y → ℂ} (hχ : ContDiff ℝ (⊤ : ℕ∞) χ)
    (hc : HasCompactSupport χ) (q : Polynomial W) (j : I) :
    ∃ C : ℝ, 0≤C ∧ ∀ δ : ℝ, 0<δ → δ≤1 →
      MemLp (cutoffLayer χ q δ j) 2 (volume.restrict halfspace) ∧
      (∫ z in halfspace, ‖cutoffLayer χ q δ j z‖^2)≤C*δ^4 := by
  sorry

end ElasticityBoundaryPacketMass

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/BoundaryPackets.lean (declaration OAI.ElasticityBoundaryPacketMass.cutoff_value_bound)

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