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

Proved

OAI.ElasticityBoundaryGradientMass.transferred_gradient_mass

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.ElasticityBoundaryGradientMass.transferred_gradient_mass 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.ElasticityBoundaryGradientMass, with open Set MeasureTheory Filter, open scoped Topology BigOperators, open ElasticityBoundaryGradientTransfer ElasticityBoundaryPhysicalTransfer):

lemma transferred_gradient_mass : ∃ C : ℝ, 0≤C ∧ ∀ (u : I → Y → ℂ),
    (∀ j, ContDiff ℝ (⊤ : ℕ∞) (u j)) →
    (∀ j, Function.support (fun z : X => u j z.ofLp)⊆K) →
    ∀ i d, (∫ x in Ω, ‖dd (transferred e η Q u i) (basis d) x‖^2)≤
      C*∑ s, ∑ α, (∫ y in H, ‖dpart (u s) α y.ofLp‖^2)

Definitions. The statement is written with OAI.ElasticityBoundaryChainRule.dd (definitions bundle OAIQ6El000 of this split); OAI.ElasticityBoundaryChartOperator.I, OAI.ElasticityBoundaryChartOperator.basis, OAI.ElasticityBoundaryGradientMass.I, OAI.ElasticityBoundaryGradientMass.X, OAI.ElasticityBoundaryGradientMass.Y, OAI.ElasticityBoundaryPhysicalTransfer.I, OAI.ElasticityBoundaryPhysicalTransfer.X, OAI.ElasticityBoundaryPhysicalTransfer.transferred and OAI.ElasticityBoundaryStrongLayer.dpart (definitions bundle OAIQ6El001 of this split).

Used by. The proof of OAI.ElasticityBoundaryPacket.Setup.field_gradient_mass of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIQ6El000
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 Topology BigOperators
namespace ElasticityBoundaryGradientMass
open ElasticityBoundaryGradientTransfer ElasticityBoundaryPhysicalTransfer
  ElasticityBoundaryChartMass ElasticityBoundaryCompactMatrix ElasticityBoundaryForceSupport
  ElasticityBoundaryFrameOperator ElasticityBoundaryRotation ElasticityBoundaryChainRule
  ElasticityBoundaryStrongLayer ElasticityBoundaryChartOperator

variable (e : OpenPartialHomeomorph X X)
    (he : ContDiffOn ℝ (⊤ : ℕ∞) e e.source)
    {K Ω H : Set X} (hK : IsCompact K) (hs : K⊆e.source)
    (hΩ : MeasurableSet Ω) (hH : MeasurableSet H)
    (hhalf : ∀ y∈e.source, e y∈Ω ↔ y∈H)
    {η : X → ℝ} (hηs : tsupport η⊆e.target) (hη : ∀ z∈e '' K, η z=1)
    (Q : X ≃ₗᵢ[ℝ] X) (Φ Ψ : X → X)
    (hΦs : ContDiff ℝ (⊤ : ℕ∞) Φ) (hΨs : ContDiff ℝ (⊤ : ℕ∞) Ψ)
    (hΦ : ∀ y∈K, Φ =ᶠ[𝓝 y] e) (hΨ : ∀ z∈e '' K, Ψ =ᶠ[𝓝 z] e.symm)

include he hK hs hΩ hH hhalf hηs hη hΦs hΨs hΦ hΨ in
lemma transferred_gradient_mass : ∃ C : ℝ, 0≤C ∧ ∀ (u : I → Y → ℂ),
    (∀ j, ContDiff ℝ (⊤ : ℕ∞) (u j)) →
    (∀ j, Function.support (fun z : X => u j z.ofLp)⊆K) →
    ∀ i d, (∫ x in Ω, ‖dd (transferred e η Q u i) (basis d) x‖^2)≤
      C*∑ s, ∑ α, (∫ y in H, ‖dpart (u s) α y.ofLp‖^2) := by
  sorry

end ElasticityBoundaryGradientMass

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.ElasticityBoundaryGradientMass.transferred_gradient_mass)

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