OAI.SKGap.actual_path_affine_tail_s2: intermediate lemma from OpenAI, Critical mixing in the Sherrington-Kirkpatrick model (result family 227)

Proved

OAI.SKGap.actual_path_affine_tail_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 MeasureTheory ProbabilityTheory Real Set, open scoped BigOperators Matrix.Norms.Frobenius NNReal ENNReal SchwartzMap):

theorem actual_path_affine_tail_s2 {j A D U W ε : ℝ}
    (hj : 0 < j) (hA : 0 < A) (hs : sqrt j*A < 1)
    (hD : 0 ≤ D) (hU : 0 ≤ U) (hW : 0 ≤ W) (hε : 0 < ε) :
    ∃ (f : 𝓢(ℝ,ℂ)) (R : ℝ) (hR : 0 ≤ R) (lo hi c : ℝ) (N : ℕ),
      R=2*sqrt j+1+1 ∧ lo=(1-sqrt j*A)^2/4 ∧ hi=2+A*(2*sqrt j+1+j*A) ∧
      0 < lo ∧ (∀ x ∈ Icc lo hi, f x=(x:ℂ)⁻¹) ∧ 0 < c ∧ 0 < N ∧
      ∀ n, N ≤ n → ∀ a : Fin n → ℝ, (∀ i, 0 ≤ a i) → (∀ i, a i ≤ A) →
      ∀ (p : Bool) (d e : Fin n → ℝ) (w : ℝ), (∀ i, |d i| ≤ D) → |w| ≤ W →
      ∀ (u v : EuclideanSpace ℝ (Fin n)), ‖u‖ ≤ U → ‖v‖ ≤ U →
      (Measure.pi (fun _ : MatrixCoordinates (Fin n) => gaussianReal 0 1)).real
        {g | ε ≤ |matrixBilinear (affineRight (pathFeature p f R hR j a) d e w
          (goeMatrix (j/(n:ℝ)) g)-diagonal (fun i =>
            (if p then (j/(n:ℝ))*∑ b,a b else 1)*d i-e i)) u v|} ≤ 2*exp (-c*(n:ℝ))

Definitions. The statement is written with OAI.SKGap.MatrixCoordinates, OAI.SKGap.affineRight, OAI.SKGap.goeMatrix, OAI.SKGap.matrixBilinear and OAI.SKGap.pathFeature (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 MeasureTheory ProbabilityTheory Real Set
open scoped BigOperators Matrix.Norms.Frobenius NNReal ENNReal SchwartzMap
variable {ι : Type*} [Fintype ι] [DecidableEq ι]

theorem actual_path_affine_tail_s2 {j A D U W ε : ℝ}
    (hj : 0 < j) (hA : 0 < A) (hs : sqrt j*A < 1)
    (hD : 0 ≤ D) (hU : 0 ≤ U) (hW : 0 ≤ W) (hε : 0 < ε) :
    ∃ (f : 𝓢(ℝ,ℂ)) (R : ℝ) (hR : 0 ≤ R) (lo hi c : ℝ) (N : ℕ),
      R=2*sqrt j+1+1 ∧ lo=(1-sqrt j*A)^2/4 ∧ hi=2+A*(2*sqrt j+1+j*A) ∧
      0 < lo ∧ (∀ x ∈ Icc lo hi, f x=(x:ℂ)⁻¹) ∧ 0 < c ∧ 0 < N ∧
      ∀ n, N ≤ n → ∀ a : Fin n → ℝ, (∀ i, 0 ≤ a i) → (∀ i, a i ≤ A) →
      ∀ (p : Bool) (d e : Fin n → ℝ) (w : ℝ), (∀ i, |d i| ≤ D) → |w| ≤ W →
      ∀ (u v : EuclideanSpace ℝ (Fin n)), ‖u‖ ≤ U → ‖v‖ ≤ U →
      (Measure.pi (fun _ : MatrixCoordinates (Fin n) => gaussianReal 0 1)).real
        {g | ε ≤ |matrixBilinear (affineRight (pathFeature p f R hR j a) d e w
          (goeMatrix (j/(n:ℝ)) g)-diagonal (fun i =>
            (if p then (j/(n:ℝ))*∑ b,a b else 1)*d i-e i)) u v|} ≤ 2*exp (-c*(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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