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

Proved

OAI.ElasticityCGO.physical_cgo_from_recursion

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.ElasticityCGO.physical_cgo_from_recursion of lean/OAI/MathematicalPhysics/Elasticity/PhysicalCGO.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.ElasticityCGO, with open MeasureTheory Set TemperedDistribution, open scoped SchwartzMap ENNReal BigOperators, open ElasticityAugmented, open ElasticityCoeffBounds (ExteriorConstant), open ElasticityRegularity (sobolevInjection), open ElasticityPhysicalAlgebra (phase physical), open ElasticityDistribution (HV Dist divd)):

theorem physical_cgo_from_recursion (R : ℝ) (hR : 1≤R)
    (Ω U : Set X) (hΩ : Ω⊆Metric.closedBall 0 R) (hU : IsOpen U) (hsub : U⊆Ω)
    (α β : X) (hα : α≠0)
    (hθ : ∑ i,((α i : ℂ)+Complex.I*(β i : ℂ))*((α i : ℂ)+Complex.I*(β i : ℂ))=0)
    (lam m : Smooth) (hl : ExteriorConstant lam) (hm : ExteriorConstant m)
    (hrl : ∀ x,star ((lam : X → ℂ) x)=(lam : X → ℂ) x)
    (hrm : ∀ x,star ((m : X → ℂ) x)=(m : X → ℂ) x)
    (hm0 : ∀ x,(m : X → ℂ) x≠0)
    (he0 : ∀ x,((lam+m+m : Smooth) : X → ℂ) x≠0)
    (χ : Fin 5 → X → ℂ) (hχ : ∀ i, ContDiff ℝ (⊤ : ℕ∞) (χ i))
    (hcχ : ∀ i,HasCompactSupport (χ i))
    (hnest : ∀ i : Fin 4,EqOn (χ i.succ) 1 (tsupport (χ i.castSucc)))
    (hs : ∀ i,tsupport (χ i)⊆Ω) (h1 : EqOn (χ 0) 1 U)
    (a : ℕ → Amplitude Smooth)
    (hR₀ : leadingR (localCoord hU) (fun i => (α i : ℂ)+Complex.I*(β i : ℂ))
      (restrictSmooth U lam) (restrictSmooth U m) (localAmp U (a 0)).1 (localAmp U (a 0)).2=0)
    (hn₀ : normal (fun i => (α i : ℂ)+Complex.I*(β i : ℂ)) (a 0).1=0)
    (hrec : ∀ j<8,RecursOn hU (fun i => (α i : ℂ)+Complex.I*(β i : ℂ)) lam m (a j) (a (j+1))) :
    ∃ C h₀ : ℝ, 0<C ∧ 0<h₀ ∧ ∀ h : ℝ,0<h → h≤h₀ →
      ∃ w : ElasticityPhysicalDual.H,∃ q : HV,∃ v : Fin 3 → S,
        (∀ i,∀ᵐ x ∂(volume : Measure X),x∈U → w i x=v i x) ∧
        (∀ i,EqOn (physical (phase h α β) lam m
          ((fun j => cut ⟨χ 0,hχ 0⟩ (hcχ 0) ((truncation (h : ℂ) 8 a).1 j))-v) i : X → ℂ) 0 U) ∧
        (∀ i,sobolevInjection 1 (q i)=smulLeftCLM ℂ (χ 0)
          ((Fin.cons (divd (phase h α β) (fun j => (w j : Dist)))
            (fun j => (w j : Dist)) : Fin 4 → Dist) i)) ∧
        ‖w‖≤C*h^7 ∧ ‖q‖≤C*h

Definitions. The statement is written with OAI.ElasticityAugmented.Amplitude, OAI.ElasticityAugmented.LocalSmooth, OAI.ElasticityAugmented.RecursOn, OAI.ElasticityAugmented.Smooth, OAI.ElasticityAugmented.X, OAI.ElasticityAugmented.leadingR, OAI.ElasticityAugmented.localAmp, OAI.ElasticityAugmented.localCoord, OAI.ElasticityAugmented.normal, OAI.ElasticityAugmented.restrictSmooth, OAI.ElasticityBessel.S, OAI.ElasticityBessel.X, OAI.ElasticityCGO.S, OAI.ElasticityDistribution.Dist, OAI.ElasticityDistribution.X, OAI.ElasticityPhysicalDual.H and OAI.ElasticityPhysicalDual.X (definitions bundle OAIQ6El000 of this split); OAI.ElasticityAugmented.truncation (definitions bundle OAIQ6El001 of this split); OAI.ElasticityRegularity.sobolevInjection (definitions bundle OAIQ6El002 of this split); OAI.ElasticityCGO.cut, OAI.ElasticityCoeffBounds.ExteriorConstant, OAI.ElasticityDistribution.H, OAI.ElasticityDistribution.HV, OAI.ElasticityDistribution.divd, OAI.ElasticityPhysicalAlgebra.phase and OAI.ElasticityPhysicalAlgebra.physical (definitions bundle OAIQ6El004 of this split).

Used by. The proofs of OAI_Elasticity_dn_common_leading_frame and OAI.Elasticity.dn_common_normal_frame of this split import it.

Preamble

import Mathlib
import Definitions.Def_OAIQ6El004

Formal statement

section

attribute [-instance] _root_.OAI.ElasticityGeneralSlicing.cylinder_temperate
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 ElasticityCGO

section
open MeasureTheory Set TemperedDistribution
open scoped SchwartzMap ENNReal BigOperators
open ElasticityAugmented
open ElasticityCoeffBounds (ExteriorConstant)
open ElasticityRegularity (sobolevInjection)
open ElasticityPhysicalAlgebra (phase physical)
open ElasticityDistribution (HV Dist divd)

theorem physical_cgo_from_recursion (R : ℝ) (hR : 1≤R)
    (Ω U : Set X) (hΩ : Ω⊆Metric.closedBall 0 R) (hU : IsOpen U) (hsub : U⊆Ω)
    (α β : X) (hα : α≠0)
    (hθ : ∑ i,((α i : ℂ)+Complex.I*(β i : ℂ))*((α i : ℂ)+Complex.I*(β i : ℂ))=0)
    (lam m : Smooth) (hl : ExteriorConstant lam) (hm : ExteriorConstant m)
    (hrl : ∀ x,star ((lam : X → ℂ) x)=(lam : X → ℂ) x)
    (hrm : ∀ x,star ((m : X → ℂ) x)=(m : X → ℂ) x)
    (hm0 : ∀ x,(m : X → ℂ) x≠0)
    (he0 : ∀ x,((lam+m+m : Smooth) : X → ℂ) x≠0)
    (χ : Fin 5 → X → ℂ) (hχ : ∀ i, ContDiff ℝ (⊤ : ℕ∞) (χ i))
    (hcχ : ∀ i,HasCompactSupport (χ i))
    (hnest : ∀ i : Fin 4,EqOn (χ i.succ) 1 (tsupport (χ i.castSucc)))
    (hs : ∀ i,tsupport (χ i)⊆Ω) (h1 : EqOn (χ 0) 1 U)
    (a : ℕ → Amplitude Smooth)
    (hR₀ : leadingR (localCoord hU) (fun i => (α i : ℂ)+Complex.I*(β i : ℂ))
      (restrictSmooth U lam) (restrictSmooth U m) (localAmp U (a 0)).1 (localAmp U (a 0)).2=0)
    (hn₀ : normal (fun i => (α i : ℂ)+Complex.I*(β i : ℂ)) (a 0).1=0)
    (hrec : ∀ j<8,RecursOn hU (fun i => (α i : ℂ)+Complex.I*(β i : ℂ)) lam m (a j) (a (j+1))) :
    ∃ C h₀ : ℝ, 0<C ∧ 0<h₀ ∧ ∀ h : ℝ,0<h → h≤h₀ →
      ∃ w : ElasticityPhysicalDual.H,∃ q : HV,∃ v : Fin 3 → S,
        (∀ i,∀ᵐ x ∂(volume : Measure X),x∈U → w i x=v i x) ∧
        (∀ i,EqOn (physical (phase h α β) lam m
          ((fun j => cut ⟨χ 0,hχ 0⟩ (hcχ 0) ((truncation (h : ℂ) 8 a).1 j))-v) i : X → ℂ) 0 U) ∧
        (∀ i,sobolevInjection 1 (q i)=smulLeftCLM ℂ (χ 0)
          ((Fin.cons (divd (phase h α β) (fun j => (w j : Dist)))
            (fun j => (w j : Dist)) : Fin 4 → Dist) i)) ∧
        ‖w‖≤C*h^7 ∧ ‖q‖≤C*h := by
  sorry
end
open Set
open ElasticityAugmented
open ElasticityNegativeOrder (cutoff)

end ElasticityCGO

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/PhysicalCGO.lean (declaration OAI.ElasticityCGO.physical_cgo_from_recursion)

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