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

Proved

OAI.DilutedSpinGlass.UniversalDictionary.physicalOverlapLimsup_bound

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.UniversalDictionary.physicalOverlapLimsup_bound of lean/OAI/Probability/DilutedSpin/DilutedSpin/PhysicalOverlapLimit.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.DilutedSpinGlass.UniversalDictionary, with open _root_.MeasureTheory ProbabilityTheory HeterogeneousMarks PhysicalRoot PrescribedTree ConcreteReservoir, open ReducedTopology Filter Set, open scoped NNReal BigOperators Topology):

lemma physicalOverlapLimsup_bound {M : Model p} {C H : ℝ}
    {Ns : ℕ → ℕ} {us : ℕ → Spec L×ℕ → ℝ} (h : FullShapeControl M C H L Ns us)
    (S : ReducedTopology) {η : ℝ} (hη : 0<η) (hη1 : η≤1) (hlarge : 8<η*(L+1:ℕ)) :
    (η/8)*physicalOverlapLimsup M C H L Ns us S η≤
      Real.sqrt (physicalShapeLimsup M C H L Ns us S (η/8))

Definitions. The statement is written with OAI.DilutedSpinGlass.ReducedTopology, OAI.DilutedSpinGlass.UniversalDictionary.FullShapeControl, OAI.DilutedSpinGlass.UniversalDictionary.Spec, OAI.DilutedSpinGlass.UniversalDictionary.physicalOverlapLimsup and OAI.DilutedSpinGlass.UniversalDictionary.physicalShapeLimsup (definitions bundle OAIQ041D000 of this split); OAI.DilutedSpinGlass.Model (the published bundle Def_DilutedSpin).

Used by. The proofs of OAI_DilutedSpinGlass_UniversalDictionary_physicalOverlapLimsup_tendsto and OAI.DilutedSpinGlass.UniversalDictionary.physicalSubtree_limsup_tendsto of this split import it.

Preamble

import Mathlib
import Definitions.Def_DilutedSpin
import Definitions.Def_OAIQ041D000

Formal statement

section

attribute [-instance] _root_.OAI.DilutedSpinGlass.PrescribedTree.upperDatumLaw_probability _root_.OAI.DilutedSpinGlass.UniversalDictionary.reservoirLaw_prob
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.UniversalDictionary
open _root_.MeasureTheory ProbabilityTheory HeterogeneousMarks PhysicalRoot PrescribedTree ConcreteReservoir
open ReducedTopology Filter Set
open scoped NNReal BigOperators Topology
variable {p L : ℕ}

lemma physicalOverlapLimsup_bound {M : Model p} {C H : ℝ}
    {Ns : ℕ → ℕ} {us : ℕ → Spec L×ℕ → ℝ} (h : FullShapeControl M C H L Ns us)
    (S : ReducedTopology) {η : ℝ} (hη : 0<η) (hη1 : η≤1) (hlarge : 8<η*(L+1:ℕ)) :
    (η/8)*physicalOverlapLimsup M C H L Ns us S η≤
      Real.sqrt (physicalShapeLimsup M C H L Ns us S (η/8)) := by
  sorry

end DilutedSpinGlass.UniversalDictionary

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/PhysicalOverlapLimit.lean (declaration OAI.DilutedSpinGlass.UniversalDictionary.physicalOverlapLimsup_bound)

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