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

Proved

OAI.ElasticityDistribution.actual_transferred_limit

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.ElasticityDistribution.actual_transferred_limit 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.ElasticityDistribution, with open Set Filter MeasureTheory TemperedDistribution, open scoped SchwartzMap Topology BigOperators, open ElasticityAugmented, open ElasticityCGO (augEmbed), open ElasticityBessel (L2), open ElasticityPhysicalEstimate (physicalAug), open ElasticityPhysicalAlgebra (phase physical)):

theorem actual_transferred_limit (B K : Set X) (α β : X)
    (hθ : ∑ i,((α i : ℂ)+Complex.I*(β i : ℂ))*((α i : ℂ)+Complex.I*(β i : ℂ))=0)
    (lam m : Coeff) (h : ℕ → ℝ) (hh : Tendsto h atTop (𝓝 0)) (hne : ∀ n,h n≠0)
    (u v : ℕ → Fin 3 → SchwartzMap X ℂ) (a : Fin 4 → SchwartzMap X ℂ)
    (hv : ∀ n i,EqOn (physical (phase (h n) α β) (eval lam) (eval m) (v n) i : X → ℂ) 0 B)
    (hs : ∀ n i,tsupport (v n i-u n i : SchwartzMap X ℂ)⊆K)
    (C : ℝ) (hC : ∀ n,‖augEmbed (physicalAug (phase (h n) α β) (v n))‖≤C)
    (ha : ∀ i,Tendsto (fun n => L2 (physicalAug (phase (h n) α β) (u n) i)) atTop (𝓝 (L2 (a i)))) :
    ∃ W : AugHilbert,
      (∀ i,LocalEq B (leadingRd (fun i => (α i : ℂ)+Complex.I*(β i : ℂ)) lam m
        (fun j => (W j.succ : Dist)) (W 0 : Dist) i) 0) ∧
      LocalEq B (leadingSd (fun i => (α i : ℂ)+Complex.I*(β i : ℂ)) lam m
        (fun j => (W j.succ : Dist)) (W 0 : Dist)) 0 ∧
      LocalEq B (normald (fun i => (α i : ℂ)+Complex.I*(β i : ℂ)) (fun j => (W j.succ : Dist))) 0 ∧
      ∀ i,LocalEq Kᶜ (W i : Dist) (a i : Dist)

Definitions. The statement is written with OAI.ElasticityAugmented.Smooth, OAI.ElasticityAugmented.X, OAI.ElasticityBessel.S, OAI.ElasticityBessel.X, OAI.ElasticityCGO.S, OAI.ElasticityDistribution.Coeff, OAI.ElasticityDistribution.Dist, OAI.ElasticityDistribution.X and OAI.ElasticityDistribution.eval (definitions bundle OAIQ6El000 of this split); OAI.ElasticityBessel.L2, OAI.ElasticityCGO.augEmbed, OAI.ElasticityDistribution.AugHilbert, OAI.ElasticityDistribution.LocalEq, OAI.ElasticityPhysicalAlgebra.phase and OAI.ElasticityPhysicalAlgebra.physical (definitions bundle OAIQ6El004 of this split); OAI.ElasticityDistribution.normald (definitions bundle OAIQ6El005 of this split); OAI.ElasticityDistribution.leadingRd, OAI.ElasticityDistribution.leadingSd 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

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 ElasticityDistribution

open Set Filter MeasureTheory TemperedDistribution
open scoped SchwartzMap Topology BigOperators
open ElasticityAugmented
open ElasticityCGO (augEmbed)
open ElasticityBessel (L2)
open ElasticityPhysicalEstimate (physicalAug)
open ElasticityPhysicalAlgebra (phase physical)

theorem actual_transferred_limit (B K : Set X) (α β : X)
    (hθ : ∑ i,((α i : ℂ)+Complex.I*(β i : ℂ))*((α i : ℂ)+Complex.I*(β i : ℂ))=0)
    (lam m : Coeff) (h : ℕ → ℝ) (hh : Tendsto h atTop (𝓝 0)) (hne : ∀ n,h n≠0)
    (u v : ℕ → Fin 3 → SchwartzMap X ℂ) (a : Fin 4 → SchwartzMap X ℂ)
    (hv : ∀ n i,EqOn (physical (phase (h n) α β) (eval lam) (eval m) (v n) i : X → ℂ) 0 B)
    (hs : ∀ n i,tsupport (v n i-u n i : SchwartzMap X ℂ)⊆K)
    (C : ℝ) (hC : ∀ n,‖augEmbed (physicalAug (phase (h n) α β) (v n))‖≤C)
    (ha : ∀ i,Tendsto (fun n => L2 (physicalAug (phase (h n) α β) (u n) i)) atTop (𝓝 (L2 (a i)))) :
    ∃ W : AugHilbert,
      (∀ i,LocalEq B (leadingRd (fun i => (α i : ℂ)+Complex.I*(β i : ℂ)) lam m
        (fun j => (W j.succ : Dist)) (W 0 : Dist) i) 0) ∧
      LocalEq B (leadingSd (fun i => (α i : ℂ)+Complex.I*(β i : ℂ)) lam m
        (fun j => (W j.succ : Dist)) (W 0 : Dist)) 0 ∧
      LocalEq B (normald (fun i => (α i : ℂ)+Complex.I*(β i : ℂ)) (fun j => (W j.succ : Dist))) 0 ∧
      ∀ i,LocalEq Kᶜ (W i : Dist) (a i : Dist) := by
  sorry

end ElasticityDistribution

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.ElasticityDistribution.actual_transferred_limit)

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