OAI.SKGap.actual_path_bias: intermediate lemma from OpenAI, A spectral gap throughout the high-temperature Sherrington-Kirkpatrick phase (result family 227)

Proved

OAI.SKGap.actual_path_bias

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.SKGap.sk_main, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, A spectral gap throughout the high-temperature Sherrington-Kirkpatrick phase (result family 227), OpenAI Math Release, September 24, 2026, result family 227, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.SKGap.actual_path_bias of lean/OAI/Probability/SKGap/Brownian/PathBias.lean at commit 1581d4c, ported to Lean 4.33.1 and Mathlib 0df444a.

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

theorem actual_path_bias {j A : ℝ} (hj : 0 < j) (hA : 0 < A) (hs : sqrt j*A < 1) :
    ∃ (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) ∧ (∀ x, star (f x)=f x) ∧
      0 < C ∧ 0 < N ∧
      (∀ (n : ℕ), N ≤ n → ∀ (a : Fin n → ℝ), (∀ i, 0 ≤ a i) →
        (∀ i, a i ≤ A) → ∀ z ∈ Icc (0:ℝ) 1, ∀ i,
          |pathExpected f R hR j a z i-1| ≤ C/(n:ℝ))

Definitions. The statement is written with OAI.SKGap.pathExpected (definitions bundle OAIQ072A000 of this split).

Used by. The proof of OAI.SKGap.actual_projected_path_equivalents 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 Filter
open RealComplex
open scoped BigOperators Matrix.Norms.Frobenius NNReal ENNReal SchwartzMap Topology

theorem actual_path_bias {j A : ℝ} (hj : 0 < j) (hA : 0 < A) (hs : sqrt j*A < 1) :
    ∃ (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) ∧ (∀ x, star (f x)=f x) ∧
      0 < C ∧ 0 < N ∧
      (∀ (n : ℕ), N ≤ n → ∀ (a : Fin n → ℝ), (∀ i, 0 ≤ a i) →
        (∀ i, a i ≤ A) → ∀ z ∈ Icc (0:ℝ) 1, ∀ i,
          |pathExpected f R hR j a z i-1| ≤ C/(n:ℝ)) := by
  sorry
end SKGap
end
end

end OAI
end

Source

OpenAI, A spectral gap throughout the high-temperature Sherrington-Kirkpatrick phase (result family 227), OpenAI Math Release, September 24, 2026, https://github.com/openai/math (result family 227; paper: https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/main.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/OAI/Probability/SKGap/Brownian/PathBias.lean (declaration OAI.SKGap.actual_path_bias)

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