OAI.ElasticityBoundaryPacket.Setup.energy_chart: intermediate lemma from OpenAI, Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem

Proved

OAI.ElasticityBoundaryPacket.Setup.energy_chart

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.ElasticityBoundaryPacket.Setup.energy_chart of lean/OAI/MathematicalPhysics/Elasticity/BoundaryEnergy.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.ElasticityBoundaryPacket, with open Set Filter MeasureTheory, open scoped Topology BigOperators, open Elasticity ElasticityBoundaryDensityPull ElasticityBoundaryPacketMass ElasticityBoundaryPhysicalMass ElasticityBoundaryCutoffLayer):

lemma Setup.energy_chart (s : Setup Ω x) (hΩ : MeasurableSet Ω) {l m : X → ℝ}
    (hl : BoundedCoefficient Ω l) (hm : BoundedCoefficient Ω m) (p q : Polynomial W) (δ : ℝ) :
    complexTrialEnergy l m (s.trial p δ) (s.trial q δ)=
      ∫ y in ElasticityBoundaryProfileBounds.halfspace,
        tensorDensity (fun i α j β => s.rawCoeff (fun z => (l z:ℂ)) (fun z => (m z:ℂ)) i α j β y)
          (fun i α => dpart (cutoffLayer s.cutoff p δ i) α y)
          (fun i α => dpart (cutoffLayer s.cutoff q δ i) α y)

Definitions. The statement is written with OAI.Elasticity.BoundedCoefficient, OAI.Elasticity.complexTrialEnergy and OAI.Elasticity.tensorDensity (definitions bundle OAIQ6El000 of this split); OAI.ElasticityBoundaryCutoffLayer.cutoffLayer, OAI.ElasticityBoundaryMVCalculus.MP, OAI.ElasticityBoundaryMVNormal.W, OAI.ElasticityBoundaryPacket.I, OAI.ElasticityBoundaryPacket.Setup, OAI.ElasticityBoundaryPacket.X, OAI.ElasticityBoundaryProfileBounds.Y, OAI.ElasticityBoundaryProfileBounds.halfspace, OAI.ElasticityBoundaryStrongLayer.dpart and OAI.ElasticityBoundaryPacket.Setup.rawCoeff (definitions bundle OAIQ6El001 of this split); OAI.ElasticityBoundaryPacket.Setup.trial (definitions bundle OAIQ6El004 of this split).

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

Preamble

import Mathlib
import Definitions.Def_OAIQ6El004

Formal statement

section

attribute [-instance] _root_.OAI.ElasticityGeneralSlicing.cylinder_temperate _root_.OAI.ElasticityODE.instFactLeRealOfNat_solutions
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 ElasticityBoundaryPacket

open Set Filter MeasureTheory
open scoped Topology BigOperators
open Elasticity ElasticityBoundaryDensityPull ElasticityBoundaryPacketMass ElasticityBoundaryPhysicalMass ElasticityBoundaryCutoffLayer
  ElasticityBoundaryStrongLayer ElasticityBoundaryMixedScaling ElasticityBoundaryGradientTransfer
  ElasticityBoundaryChartOperator ElasticityBoundaryMVNormal ElasticityBoundaryPhysicalTransfer
  ElasticityBoundaryCutoffParametrix ElasticityBoundaryActualTower

variable {Ω : Set X} {x : X}

lemma Setup.energy_chart (s : Setup Ω x) (hΩ : MeasurableSet Ω) {l m : X → ℝ}
    (hl : BoundedCoefficient Ω l) (hm : BoundedCoefficient Ω m) (p q : Polynomial W) (δ : ℝ) :
    complexTrialEnergy l m (s.trial p δ) (s.trial q δ)=
      ∫ y in ElasticityBoundaryProfileBounds.halfspace,
        tensorDensity (fun i α j β => s.rawCoeff (fun z => (l z:ℂ)) (fun z => (m z:ℂ)) i α j β y)
          (fun i α => dpart (cutoffLayer s.cutoff p δ i) α y)
          (fun i α => dpart (cutoffLayer s.cutoff q δ i) α y) := by
  sorry

end ElasticityBoundaryPacket

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/BoundaryEnergy.lean (declaration OAI.ElasticityBoundaryPacket.Setup.energy_chart)

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