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

Proved

OAI.ElasticityPhysicalEstimate.physical_compact_comparison

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.ElasticityPhysicalEstimate.physical_compact_comparison of lean/OAI/MathematicalPhysics/Elasticity/Transfer.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.ElasticityPhysicalEstimate, with open Set, open scoped BigOperators SchwartzMap, open ElasticityBessel ElasticitySchwartzCarleman ElasticityNegativeOrder ElasticityPhysicalAlgebra, open ElasticityAugmented (coord coeffA coeffV), open ElasticityCoeffBounds):

theorem physical_compact_comparison {B K : Set X} (hB : IsOpen B)
    (hK : IsClosed K) (hKB : K⊆B) (R : ℝ) (hR : 1≤R)
    (hKR : K⊆Metric.closedBall 0 R) (α β : X) (hα : α≠0)
    (lam m : Fin 2 → Smooth)
    (hl : ∀ q,ExteriorConstant (lam q)) (hm : ∀ q,ExteriorConstant (m q))
    (hm0 : ∀ q x,(m q : X → ℂ) x≠0)
    (he0 : ∀ q x,((lam q+m q+m q : Smooth) : X → ℂ) x≠0)
    (hel : EqOn (lam 0 : X → ℂ) (lam 1) Kᶜ)
    (hem : EqOn (m 0 : X → ℂ) (m 1) Kᶜ) :
    ∃ C h₀ : ℝ,0<C ∧ 0<h₀ ∧ ∀ h : ℝ,0<h → h≤h₀ →
      ∀ u v : Fin 3 → S,
      (∀ i,tsupport ((v-u) i : X → ℂ)⊆K) →
      (∀ i,EqOn (physical (phase h α β) (lam 0) (m 0) u i : X → ℂ) 0 B) →
      (∀ i,EqOn (physical (phase h α β) (lam 1) (m 1) v i : X → ℂ) 0 B) →
      sobNorm h 1 (physicalAug (phase h α β) (v-u)) ≤
        C*sobNorm h 1 (physicalAug (phase h α β) u)

Definitions. The statement is written with OAI.ElasticityAugmented.X, OAI.ElasticityBessel.S, OAI.ElasticityBessel.X and OAI.ElasticityPhysicalAlgebra.Smooth (definitions bundle OAIQ6El000 of this split); OAI.ElasticityCoeffBounds.ExteriorConstant, OAI.ElasticityPhysicalAlgebra.phase and OAI.ElasticityPhysicalAlgebra.physical (definitions bundle OAIQ6El004 of this split); OAI.ElasticityPhysicalEstimate.X (definitions bundle OAIQ6El005 of this split); OAI.ElasticityNegativeOrder.sobNorm and OAI.ElasticityPhysicalEstimate.physicalAug (definitions bundle OAIQ6El006 of this split).

Used by. The proof of OAI.Elasticity.dn_weak_transport_column of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIQ6El006

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

section
noncomputable section
open Set
open scoped BigOperators SchwartzMap
namespace ElasticityPhysicalEstimate
open ElasticityBessel ElasticitySchwartzCarleman ElasticityNegativeOrder ElasticityPhysicalAlgebra
open ElasticityAugmented (coord coeffA coeffV)
open ElasticityCoeffBounds

theorem physical_compact_comparison {B K : Set X} (hB : IsOpen B)
    (hK : IsClosed K) (hKB : K⊆B) (R : ℝ) (hR : 1≤R)
    (hKR : K⊆Metric.closedBall 0 R) (α β : X) (hα : α≠0)
    (lam m : Fin 2 → Smooth)
    (hl : ∀ q,ExteriorConstant (lam q)) (hm : ∀ q,ExteriorConstant (m q))
    (hm0 : ∀ q x,(m q : X → ℂ) x≠0)
    (he0 : ∀ q x,((lam q+m q+m q : Smooth) : X → ℂ) x≠0)
    (hel : EqOn (lam 0 : X → ℂ) (lam 1) Kᶜ)
    (hem : EqOn (m 0 : X → ℂ) (m 1) Kᶜ) :
    ∃ C h₀ : ℝ,0<C ∧ 0<h₀ ∧ ∀ h : ℝ,0<h → h≤h₀ →
      ∀ u v : Fin 3 → S,
      (∀ i,tsupport ((v-u) i : X → ℂ)⊆K) →
      (∀ i,EqOn (physical (phase h α β) (lam 0) (m 0) u i : X → ℂ) 0 B) →
      (∀ i,EqOn (physical (phase h α β) (lam 1) (m 1) v i : X → ℂ) 0 B) →
      sobNorm h 1 (physicalAug (phase h α β) (v-u)) ≤
        C*sobNorm h 1 (physicalAug (phase h α β) u) := by
  sorry
end ElasticityPhysicalEstimate

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/Transfer.lean (declaration OAI.ElasticityPhysicalEstimate.physical_compact_comparison)

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