OAI.DilutedSpinGlass.PrescribedTree.finiteTrial_upper_fixedField: intermediate lemma from OpenAI, The Mézard–Parisi formula for diluted spin glasses (result family 221)

Proved

OAI.DilutedSpinGlass.PrescribedTree.finiteTrial_upper_fixedField

by dbenbenn · Oct 27, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.DilutedSpinGlass.mezard_parisi, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, The Mézard–Parisi formula for diluted spin glasses (result family 221), OpenAI Math Release, September 23, 2026, result family 221, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.DilutedSpinGlass.PrescribedTree.finiteTrial_upper_fixedField of lean/OAI/Probability/DilutedSpin/DilutedSpin/CavityGapFubini.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.DilutedSpinGlass.PrescribedTree, with open _root_.MeasureTheory ProbabilityTheory KernelTower, open scoped BigOperators NNReal):

lemma finiteTrial_upper_fixedField (M : Model (q+1)) (hM : Admissible M)
    (a : Λ) (r : ℕ) (Q : FiniteLaw R) (U : R → KernelTower Λ r)
    (x : R → FinitePath Λ r → ℝ) (m : Fin r → ℝ) (hm : Exponents m) (h : Fin N → ℝ) :
    (∫ k,∫ z : Fin k → InteractionSample (q+1),
      (FiniteLaw.uniform : FiniteLaw (Fin k → Fin (q+1) → Fin N)).expect
        (fun i => logPartition z h i-(N:ℝ)*Real.log 2)
      ∂Measure.pi (fun _ : Fin k => M.disorder.toMeasure) ∂poissonMeasure (M.alpha*N))+
      (M.alpha:ℝ)*N*q*(∫ z,trialLog r (finiteTrialLaw r Q U x) m
        (fun y => Real.log (edge z.1 y)) ∂M.disorder.toMeasure) ≤
      ∑ s : Fin N,∫ k,∫ z : Fin k → InteractionSample (q+1),
        trialLog r (finiteTrialLaw r Q U x) m (siteLog z (h s))
        ∂Measure.pi (fun _ : Fin k => M.disorder.toMeasure) ∂poissonMeasure (M.alpha*(q+1))

Definitions. The statement is written with OAI.DilutedSpinGlass.FiniteLaw, OAI.DilutedSpinGlass.FinitePath, OAI.DilutedSpinGlass.KernelTower, OAI.DilutedSpinGlass.FiniteLaw.expect, OAI.DilutedSpinGlass.FiniteLaw.uniform and OAI.DilutedSpinGlass.PrescribedTree.finiteTrialLaw (definitions bundle OAIQ041D000 of this split); OAI.DilutedSpinGlass.Admissible, OAI.DilutedSpinGlass.Exponents, OAI.DilutedSpinGlass.Interaction, OAI.DilutedSpinGlass.InteractionSample, OAI.DilutedSpinGlass.Model, OAI.DilutedSpinGlass.Spin, OAI.DilutedSpinGlass.edge, OAI.DilutedSpinGlass.logPartition, OAI.DilutedSpinGlass.siteLog and OAI.DilutedSpinGlass.trialLog (the published bundle Def_DilutedSpin).

Used by. The proof of OAI.DilutedSpinGlass.mezard_parisi of this split imports it.

Preamble

import Mathlib
import Definitions.Def_DilutedSpin
import Definitions.Def_OAIQ041D000

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
namespace DilutedSpinGlass.PrescribedTree
open _root_.MeasureTheory ProbabilityTheory KernelTower
open scoped BigOperators NNReal
variable {Λ R : Type} [Fintype Λ] [Fintype R]
    [MeasurableSpace R] [MeasurableSingletonClass R] {N q : ℕ} [NeZero N]

lemma finiteTrial_upper_fixedField (M : Model (q+1)) (hM : Admissible M)
    (a : Λ) (r : ℕ) (Q : FiniteLaw R) (U : R → KernelTower Λ r)
    (x : R → FinitePath Λ r → ℝ) (m : Fin r → ℝ) (hm : Exponents m) (h : Fin N → ℝ) :
    (∫ k,∫ z : Fin k → InteractionSample (q+1),
      (FiniteLaw.uniform : FiniteLaw (Fin k → Fin (q+1) → Fin N)).expect
        (fun i => logPartition z h i-(N:ℝ)*Real.log 2)
      ∂Measure.pi (fun _ : Fin k => M.disorder.toMeasure) ∂poissonMeasure (M.alpha*N))+
      (M.alpha:ℝ)*N*q*(∫ z,trialLog r (finiteTrialLaw r Q U x) m
        (fun y => Real.log (edge z.1 y)) ∂M.disorder.toMeasure) ≤
      ∑ s : Fin N,∫ k,∫ z : Fin k → InteractionSample (q+1),
        trialLog r (finiteTrialLaw r Q U x) m (siteLog z (h s))
        ∂Measure.pi (fun _ : Fin k => M.disorder.toMeasure) ∂poissonMeasure (M.alpha*(q+1)) := by
  sorry

end DilutedSpinGlass.PrescribedTree

end

end OAI
end

Source

OpenAI, The Mézard–Parisi formula for diluted spin glasses (result family 221), OpenAI Math Release, September 23, 2026, https://github.com/openai/math (result family 221; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Mezard-Parisi-formula-for-diluted-spin-glasses-September-23-2026/paper.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/DilutedSpin/DilutedSpin/CavityGapFubini.lean (declaration OAI.DilutedSpinGlass.PrescribedTree.finiteTrial_upper_fixedField)

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