OAI.CriticalSK.critical_two_clock_mixing_s2: intermediate lemma from OpenAI, Critical mixing in the Sherrington-Kirkpatrick model (result family 227)

Proved

OAI.CriticalSK.critical_two_clock_mixing_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); in Lean it is the declaration OAI.CriticalSK.critical_two_clock_mixing of lean/OAI/MathematicalPhysics/CriticalMixing/Main.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.CriticalSK, with open scoped BigOperators Topology NNReal ENNReal, open MeasureTheory ProbabilityTheory Filter):

theorem critical_two_clock_mixing_s2 (ε : ℝ) (hε : 0 < ε) :
    Tendsto (fun n : ℕ => (disorderLaw n).real (twoClockMixingEvent n ε))
      atTop (𝓝 1)

Definitions. The statement is written with OAI.CriticalSK.twoClockMixingEvent (definitions bundle OAIQ074C000 of this split); OAI.CriticalSK.Disorder, OAI.CriticalSK.Edge and OAI.CriticalSK.disorderLaw (the published bundle Def_CriticalSKMixing).

Used by. The proof of OAI.CriticalSK.critical_mixing_bounds of this split imports it.

Preamble

import Mathlib
import Definitions.Def_CriticalSKMixing
import Definitions.Def_OAIQ074C000

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
open scoped BigOperators Topology NNReal ENNReal
open MeasureTheory ProbabilityTheory Filter

namespace CriticalSK

theorem critical_two_clock_mixing_s2 (ε : ℝ) (hε : 0 < ε) :
    Tendsto (fun n : ℕ => (disorderLaw n).real (twoClockMixingEvent n ε))
      atTop (𝓝 1) := by
  sorry

end CriticalSK
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); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/MathematicalPhysics/CriticalMixing/Main.lean (declaration OAI.CriticalSK.critical_two_clock_mixing)

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