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

Proved

OAI.ElasticityGeneralSlicing.cylinder_weak_slices

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.ElasticityGeneralSlicing.cylinder_weak_slices of lean/OAI/MathematicalPhysics/Elasticity/WeakLimits.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.ElasticityGeneralSlicing, with open Set MeasureTheory TemperedDistribution, open scoped SchwartzMap ENNReal LineDeriv, open ElasticityPlanar):

theorem cylinder_weak_slices (u w : Lp ℂ 2 (volume : Measure (P × ℂ)))
    (hu : Integrable (u : P × ℂ → ℂ)) (hw : Integrable (w : P × ℂ → ℂ))
    (he : ∂_{((0:P),(1:ℂ))} (u : 𝓢'(P × ℂ,ℂ))+
      Complex.I • ∂_{((0:P),Complex.I)} (u : 𝓢'(P × ℂ,ℂ))+(w : 𝓢'(P × ℂ,ℂ))=0) :
    ∀ᵐ t : P, MemLp (fun z => u (t,z)) 2 volume ∧
      MemLp (fun z => w (t,z)) 2 volume ∧
      ∀ φ : Test, (∫ z, φ z*w (t,z))=(∫ z, testDbar φ z*u (t,z))

Definitions. The statement is written with OAI.ElasticityGeneralSlicing.cylinder_temperate, OAI.ElasticityGeneralSlicing.testDbar and OAI.ElasticityPlanar.Test (definitions bundle OAIQ6El002 of this split).

Used by. The proofs of OAI_ElasticityGeneralSlicing_coupled_slice_equations, OAI.ElasticityGeneralSlicing.smooth_cylinder_representative and OAI.ElasticityGeneralSlicing.smooth_supported_of_cylinder of this split import it.

Preamble

import Mathlib
import Definitions.Def_OAIQ6El002

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

noncomputable section

namespace ElasticityGeneralSlicing
open Set MeasureTheory TemperedDistribution
open scoped SchwartzMap ENNReal LineDeriv
variable {P : Type*} [NormedAddCommGroup P] [InnerProductSpace ℝ P]
    [FiniteDimensional ℝ P] [MeasureSpace P] [BorelSpace P]
    [Measure.IsAddHaarMeasure (volume : Measure P)]
open ElasticityPlanar

theorem cylinder_weak_slices (u w : Lp ℂ 2 (volume : Measure (P × ℂ)))
    (hu : Integrable (u : P × ℂ → ℂ)) (hw : Integrable (w : P × ℂ → ℂ))
    (he : ∂_{((0:P),(1:ℂ))} (u : 𝓢'(P × ℂ,ℂ))+
      Complex.I • ∂_{((0:P),Complex.I)} (u : 𝓢'(P × ℂ,ℂ))+(w : 𝓢'(P × ℂ,ℂ))=0) :
    ∀ᵐ t : P, MemLp (fun z => u (t,z)) 2 volume ∧
      MemLp (fun z => w (t,z)) 2 volume ∧
      ∀ φ : Test, (∫ z, φ z*w (t,z))=(∫ z, testDbar φ z*u (t,z)) := by
  sorry

end ElasticityGeneralSlicing

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/WeakLimits.lean (declaration OAI.ElasticityGeneralSlicing.cylinder_weak_slices)

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