OAI.SKGap.actualWord_expected_recursion_s2: intermediate lemma from OpenAI, Critical mixing in the Sherrington-Kirkpatrick model (result family 227)

Proved

OAI.SKGap.actualWord_expected_recursion_s2

by dbenbenn · Oct 28, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.CriticalSK.critical_mixing_bounds, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, Critical mixing in the Sherrington-Kirkpatrick model (result family 227), OpenAI Math Release, September 25, 2026, result family 227, github.com/openai/math, Apache License 2.0), ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.SKGap, with open Matrix Real MeasureTheory ProbabilityTheory Set, open scoped BigOperators Matrix.Norms.Frobenius SchwartzMap):

theorem actualWord_expected_recursion_s2 {n : ℕ} (hn : 0<n) (f : 𝓢(ℝ,ℂ))
    {j A D z : ℝ} (hj : 0<j) (hA : 0<A) (hD : 0≤D) (hs : sqrt j*A<1)
    (hf : ∀ x∈Icc ((1-sqrt j*A)^2/4) (2+A*(2*sqrt j+1+j*A)),f x=(x:ℂ)⁻¹)
    (herr : 2*sqrt A*sqrt (sqrt (2*j^2*A^2/(n:ℝ)))≤(1-sqrt j*A)^2/4)
    {a : Fin n→ℝ} (ha : ∀ i,0≤a i) (haA : ∀ i,a i≤A) (haD : ∀ i,|a i|≤D)
    (hz : z∈Icc (0:ℝ) 1) (P Q : List (WordLetter (Fin n)))
    (hP : ∀ l∈P,l.bounded D) (hQ : ∀ l∈Q,l.bounded D) (i : Fin n) :
    let R := 2*sqrt j+1+1
    let hR : 0≤R := by dsimp [R];positivity
    let B : NNReal := ⟨actualWordBound f R j A D,(actualWordBound_pos f hR hj.le hA.le hD).le⟩
    |wordExpected f hR j a z (P++(.noise::Q)) i-wordRecursionAtExpected f hR j a z P Q i|≤
      (wordMarkedCost B j R A (pathRate j A) z P.length Q.length+
        wordRecursionAtCost j B z a P Q)/(n:ℝ)

Definitions. The statement is written with OAI.SKGap.WordLetter, OAI.SKGap.actualWordBound, OAI.SKGap.actualWordBound_pos, OAI.SKGap.pathRate, OAI.SKGap.wordExpected, OAI.SKGap.wordMarkedCost, OAI.SKGap.wordRecursionAtCost, OAI.SKGap.wordRecursionAtExpected and OAI.SKGap.WordLetter.bounded (definitions bundle OAIQ072A000 of this split).

Used by. No other item of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIQ072A000

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

section
noncomputable section
namespace SKGap
open Matrix Real MeasureTheory ProbabilityTheory Set
open scoped BigOperators Matrix.Norms.Frobenius SchwartzMap

theorem actualWord_expected_recursion_s2 {n : ℕ} (hn : 0<n) (f : 𝓢(ℝ,ℂ))
    {j A D z : ℝ} (hj : 0<j) (hA : 0<A) (hD : 0≤D) (hs : sqrt j*A<1)
    (hf : ∀ x∈Icc ((1-sqrt j*A)^2/4) (2+A*(2*sqrt j+1+j*A)),f x=(x:ℂ)⁻¹)
    (herr : 2*sqrt A*sqrt (sqrt (2*j^2*A^2/(n:ℝ)))≤(1-sqrt j*A)^2/4)
    {a : Fin n→ℝ} (ha : ∀ i,0≤a i) (haA : ∀ i,a i≤A) (haD : ∀ i,|a i|≤D)
    (hz : z∈Icc (0:ℝ) 1) (P Q : List (WordLetter (Fin n)))
    (hP : ∀ l∈P,l.bounded D) (hQ : ∀ l∈Q,l.bounded D) (i : Fin n) :
    let R := 2*sqrt j+1+1
    let hR : 0≤R := by dsimp [R];positivity
    let B : NNReal := ⟨actualWordBound f R j A D,(actualWordBound_pos f hR hj.le hA.le hD).le⟩
    |wordExpected f hR j a z (P++(.noise::Q)) i-wordRecursionAtExpected f hR j a z P Q i|≤
      (wordMarkedCost B j R A (pathRate j A) z P.length Q.length+
        wordRecursionAtCost j B z a P Q)/(n:ℝ) := by
  sorry
end SKGap
end
end

end OAI
end

Source

OpenAI, Critical mixing in the Sherrington-Kirkpatrick model (result family 227), OpenAI Math Release, September 25, 2026, https://github.com/openai/math (result family 227; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Critical-mixing-in-the-Sherrington-Kirkpatrick-model-September-25-2026/paper.pdf; Apache License 2.0)

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