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

Proved

OAI.ElasticityAugmented.exists_physical_recursion_step

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.ElasticityAugmented.exists_physical_recursion_step of lean/OAI/MathematicalPhysics/Elasticity/CGO.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.ElasticityAugmented, with open Set, open scoped BigOperators):

theorem exists_physical_recursion_step {U : Set X} (hU : IsOpen U) (hC : IsCompact (closure U))
    (θ : Fin 3 → ℂ) (hθ : ∑ i,θ i*θ i=0) (hne : θ≠0)
    (k r : Smooth) (hr : ∀ x,(r : X → ℂ) x≠0) (hk : ∀ x,(k : X → ℂ) x≠0)
    (hl : ∀ x,((k+r*r : Smooth) : X → ℂ) x≠0)
    (a : Amplitude Smooth) (ha : CompatibleOn hU θ (k-r*r) (r*r) a) :
    ∃ b : Amplitude Smooth, RecursOn hU θ (k-r*r) (r*r) a b ∧
      CompatibleOn hU θ (k-r*r) (r*r) b

Definitions. The statement is written with OAI.ElasticityAugmented.Amplitude, OAI.ElasticityAugmented.CompatibleOn, OAI.ElasticityAugmented.RecursOn, OAI.ElasticityAugmented.Smooth and OAI.ElasticityAugmented.X (definitions bundle OAIQ6El000 of this split).

Used by. The proof of OAI.ElasticityAugmented.exists_real_physical_series of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIQ6El000

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 ElasticityAugmented
open Set
open scoped BigOperators

theorem exists_physical_recursion_step {U : Set X} (hU : IsOpen U) (hC : IsCompact (closure U))
    (θ : Fin 3 → ℂ) (hθ : ∑ i,θ i*θ i=0) (hne : θ≠0)
    (k r : Smooth) (hr : ∀ x,(r : X → ℂ) x≠0) (hk : ∀ x,(k : X → ℂ) x≠0)
    (hl : ∀ x,((k+r*r : Smooth) : X → ℂ) x≠0)
    (a : Amplitude Smooth) (ha : CompatibleOn hU θ (k-r*r) (r*r) a) :
    ∃ b : Amplitude Smooth, RecursOn hU θ (k-r*r) (r*r) a b ∧
      CompatibleOn hU θ (k-r*r) (r*r) b := by
  sorry
end ElasticityAugmented

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/CGO.lean (declaration OAI.ElasticityAugmented.exists_physical_recursion_step)

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