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

Proved

OAI.SKGap.FullMain.quenched_posterior_covariance

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.FullMain.quenched_posterior_covariance of lean/OAI/Probability/SKGap/Posterior/QuenchedPosterior.lean at commit 1581d4c, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.SKGap.FullMain, with open MeasureTheory ProbabilityTheory Filter Set Real, open scoped Topology BigOperators, open SKGapCutoff SKGapCutoff.Recipe SKGapCutoff.Primary SKGapCutoff.Static):

theorem quenched_posterior_covariance {β Aroot ε c r₀ : ℝ}
    (hβ : 0<β) (hβ1 : β<1) (hAroot : 1<Aroot) (hs : β*Aroot<1)
    (hε : 0<ε) (hc : 0 < c) (hr₀ : 0 < r₀) :
    ∃D : ℝ,0≤D ∧ Whp β (fun n g=>PosteriorCovarianceProperty (n := n) (β^2) Aroot ε c r₀ D g)

Definitions. The statement is written with OAI.SKGap.FullMain.PosteriorCovarianceProperty and OAI.SKGap.FullMain.Whp (definitions bundle OAIQ072A000 of this split); OAI.SKGap.Disorder (the published bundle Def_SKHighTemperature).

Used by. The proof of OAI.SKGap.sk_main of this split imports it.

Preamble

import Mathlib
import Definitions.Def_SKHighTemperature
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

namespace SKGap.FullMain
open MeasureTheory ProbabilityTheory Filter Set Real
open scoped Topology BigOperators
open SKGapCutoff SKGapCutoff.Recipe SKGapCutoff.Primary SKGapCutoff.Static
noncomputable section

theorem quenched_posterior_covariance {β Aroot ε c r₀ : ℝ}
    (hβ : 0<β) (hβ1 : β<1) (hAroot : 1<Aroot) (hs : β*Aroot<1)
    (hε : 0<ε) (hc : 0 < c) (hr₀ : 0 < r₀) :
    ∃D : ℝ,0≤D ∧ Whp β (fun n g=>PosteriorCovarianceProperty (n := n) (β^2) Aroot ε c r₀ D g) := by
  sorry

end
end SKGap.FullMain

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/Posterior/QuenchedPosterior.lean (declaration OAI.SKGap.FullMain.quenched_posterior_covariance)

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