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

Proved

OAI.DilutedSpinGlass.PhysicalRoot.spin_selected_concentration

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.PhysicalRoot.spin_selected_concentration of lean/OAI/Probability/DilutedSpin/DilutedSpin/PhysicalSelectedConcentration.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.DilutedSpinGlass.PhysicalRoot, with open _root_.MeasureTheory ProbabilityTheory HeterogeneousMarks Set, open scoped NNReal ENNReal BigOperators):

theorem spin_selected_concentration
    (μ : Measure X) [IsProbabilityMeasure μ] (ν : Measure I) [IsProbabilityMeasure ν]
    (ξ : Measure Y) [IsProbabilityMeasure ξ] (r s : ℝ≥0)
    (V : X → (Fin N → Spin) → ℝ) (field : Y → ℝ)
    (hVm : ∀ σ, Measurable (fun x => V x σ)) (hhm : Measurable field)
    {C H c : ℝ} (hC : 0 ≤ C) (hc : 0 < c)
    (hV : ∀ x σ, |V x σ| ≤ C) (hh : ∀ y, |field y| ≤ H)
    (Q : (i : I) → Fin (L+1) → FiniteLaw (A i)) (m : Fin (L+1) → ℝ)
    (hm : ∀ j, c ≤ m j)
    (sel : I → Bool)
    (fixed D E : (i : I) → FinitePath (Fin N → Spin) (L+1) → FinitePath (A i) (L+1) → ℝ)
    (hf : ∀ i x y, |Real.log (fixed i x y)| ≤ 1)
    (hD : ∀ i x y, |D i x y| ≤ 1) (hE : ∀ i x y, |E i x y| ≤ 1)
    {t a b ρ : ℝ} (ht : |t| ≤ 1/4) (hab : a < b) (hρ : 0 < ρ)
    (hJ : Icc (a-ρ) (b+ρ) ⊆ Ioo (-(1:ℝ)/4) (1/4)) :
    let T := KernelTower.terminalTower (fun _ : Fin N => false) FiniteLaw.uniform L
    let base := fun h k (x : RootPath X k) z => energy V field h k x (KernelTower.terminalState L z)
    (∫ u in a..b, fullSelectedError (fun _ : Fin N => ξ) μ ν r s T Q m base sel fixed D E t u)/(b-a) ≤
      Real.sqrt ((b-a)*(4+4*(b-a))/c*(s:ℝ))/(b-a)+
      4*Real.sqrt (3*C^2*(r:ℝ)+4*(s:ℝ)+2*H^2*N)/ρ+
      12*ρ*(s:ℝ)/(b-a)+Real.sqrt s

Definitions. The statement is written with OAI.DilutedSpinGlass.FiniteLaw, OAI.DilutedSpinGlass.FinitePath, OAI.DilutedSpinGlass.KernelTower, OAI.DilutedSpinGlass.RootPath, OAI.DilutedSpinGlass.FiniteLaw.uniform, OAI.DilutedSpinGlass.HeterogeneousMarks.fullSelectedError, OAI.DilutedSpinGlass.KernelTower.terminalState, OAI.DilutedSpinGlass.KernelTower.terminalTower and OAI.DilutedSpinGlass.PhysicalRoot.energy (definitions bundle OAIQ041D000 of this split); OAI.DilutedSpinGlass.Spin (the published bundle Def_DilutedSpin).

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

Preamble

import Mathlib
import Definitions.Def_DilutedSpin
import Definitions.Def_OAIQ041D000

Formal statement

section

attribute [-instance] _root_.OAI.DilutedSpinGlass.ConcreteReservoir.bondLaw_probability _root_.OAI.DilutedSpinGlass.ConcreteReservoir.siteNeZero _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
section
namespace DilutedSpinGlass.PhysicalRoot
open _root_.MeasureTheory ProbabilityTheory HeterogeneousMarks Set
open scoped NNReal ENNReal BigOperators
variable {X Y I : Type} [MeasurableSpace X] [MeasurableSpace Y]
  [Countable I] [MeasurableSpace I] [MeasurableSingletonClass I]
  {A : I → Type} [∀ i, Fintype (A i)] {N L : ℕ}

theorem spin_selected_concentration
    (μ : Measure X) [IsProbabilityMeasure μ] (ν : Measure I) [IsProbabilityMeasure ν]
    (ξ : Measure Y) [IsProbabilityMeasure ξ] (r s : ℝ≥0)
    (V : X → (Fin N → Spin) → ℝ) (field : Y → ℝ)
    (hVm : ∀ σ, Measurable (fun x => V x σ)) (hhm : Measurable field)
    {C H c : ℝ} (hC : 0 ≤ C) (hc : 0 < c)
    (hV : ∀ x σ, |V x σ| ≤ C) (hh : ∀ y, |field y| ≤ H)
    (Q : (i : I) → Fin (L+1) → FiniteLaw (A i)) (m : Fin (L+1) → ℝ)
    (hm : ∀ j, c ≤ m j)
    (sel : I → Bool)
    (fixed D E : (i : I) → FinitePath (Fin N → Spin) (L+1) → FinitePath (A i) (L+1) → ℝ)
    (hf : ∀ i x y, |Real.log (fixed i x y)| ≤ 1)
    (hD : ∀ i x y, |D i x y| ≤ 1) (hE : ∀ i x y, |E i x y| ≤ 1)
    {t a b ρ : ℝ} (ht : |t| ≤ 1/4) (hab : a < b) (hρ : 0 < ρ)
    (hJ : Icc (a-ρ) (b+ρ) ⊆ Ioo (-(1:ℝ)/4) (1/4)) :
    let T := KernelTower.terminalTower (fun _ : Fin N => false) FiniteLaw.uniform L
    let base := fun h k (x : RootPath X k) z => energy V field h k x (KernelTower.terminalState L z)
    (∫ u in a..b, fullSelectedError (fun _ : Fin N => ξ) μ ν r s T Q m base sel fixed D E t u)/(b-a) ≤
      Real.sqrt ((b-a)*(4+4*(b-a))/c*(s:ℝ))/(b-a)+
      4*Real.sqrt (3*C^2*(r:ℝ)+4*(s:ℝ)+2*H^2*N)/ρ+
      12*ρ*(s:ℝ)/(b-a)+Real.sqrt s := by
  sorry

end DilutedSpinGlass.PhysicalRoot
end

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/PhysicalSelectedConcentration.lean (declaration OAI.DilutedSpinGlass.PhysicalRoot.spin_selected_concentration)

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