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

Proved

OAI.DilutedSpinGlass.UniversalDictionary.physical_grid_scheduled_step

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.physical_grid_scheduled_step of lean/OAI/Probability/DilutedSpin/DilutedSpin/PhysicalScheduledStep.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, open scoped NNReal BigOperators):

theorem physical_grid_scheduled_step (M : Model p) (θB hB : ℝ) (N H r d : ℕ)
    (u : Spec (H+1+1+r+1+d) × ℕ → ℝ)
    (k : ℕ+) (hk : 2≤(k:ℕ)) (a : α) (C : Fin k → ReducedTopology)
    (e : (j : Fin k) → (C j).Vertex → {j : α // j≠a})
    (Q : α → Fin (H+1+1+r+1+(d+1))) (ha : (Q a).val=r+1+(d+1))
    (had : ∀ j, ReducedTopology.Admissible (C j) (fun v => (Q (e j v)).val) (r+1+(d+1)+1) (r+1+(d+1)+1+H))
    {η : ℝ} (hlarge : 1<η*(H+1+1+r+1+(d+1):ℕ)) (hr : DepthAverage.Regular η Q) :
    let L₀ := H+1+1+r+1+d
    let μ := fullRootLaw (fun _ : Fin N => M.field.toMeasure) (bondLaw M N)
      (markLaw (weights L₀) N) (M.alpha*N) (scoreRate N)
    let K := rootTower (KernelTower.terminalTower (fun _ : Fin N => false) FiniteLaw.uniform L₀)
      (fun i : Labels L₀ (Site N) => prior i.1.1) (gridExponents L₀) (physicalBase M θB hB N)
      (dictionaryFactor (observableAt direction N) (observableAt anchor N) u)
    let f := rootVector (readVector (fun v x => readSpin (KernelTower.terminalState L₀ x) v))
      (I := Labels L₀ (Site N)) (A := fun i => Alphabet i.1.1) (X := Bond p N) (Y := ℝ) (M := N)
    let ht := shiftedFrameHeight H r (d+1)
    let OldChild := fun j => realize (H+1) (r+1+(d+1)+1) (C j) (fun v => (Q (e j v)).val)
    let NewChild := fun j => realize H (r+1+1+(d+1)+1) (C j) (fun v => (Q (e j v)).val+1)
    let OldNode := PrescribedTree.node k OldChild
    let NewNode := PrescribedTree.node k NewChild
    let BaseNode := PrescribedTree.node k (fun j => realize H (r+1+(d+1)+1) (C j) (fun v => (Q (e j v)).val))
    let B := shiftedPrefixDepths (stem BaseNode r) (d+1)
    let S := splitFrame OldNode r (d+1)
    let RawTarget := splitFrame NewNode (r+1) (d+1)
    let Target := heightCast ht RawTarget
    let _ := fun z => kernelHeightCast ht.symm (K z)
    let _ := fun z => vectorHeightCast ht.symm (f z)
    let W := (k:ℝ)*∑ j, Real.sqrt (∫ z, scheduledShapeEnergy (L₀+1) (r+1+(d+1)) (C j) (fun v => (Q (e j v)).val) (K z) (f z) ∂μ)
    let W' := (k:ℝ)*∑ j, Real.sqrt (∫ z, scheduledShapeEnergy (L₀+1) (r+1+1+(d+1)) (C j) (fun v => (Q (e j v)).val+1) (K z) (f z) ∂μ)
    let h := fun z => projectionShiftError a C e (K z) (f z) Q
    ∀ (b : OldNode.Leaf) (b' : NewNode.Leaf) (q0 : Option (Fin (t+1)) → RawTarget.Leaf),
      Function.Bijective q0 → q0 none=splitFrameLeaf NewNode (r+1) (d+1) 0 b' →
        q0 (some (Fin.last t))=splitFrameLeaf NewNode (r+1) (d+1) 1 b' →
    let v := some (Fin.last t)
    let cs := canonicalMatrixTail t
    let q := fun j => leafHeightCast ht RawTarget (q0 j)
    let x := splitFrameLeaf OldNode r (d+1) 0 b
    let m := Fin.cons 0 (gridExponents L₀)
    let J := partialKappa Target m (Finset.univ.image q)/
      partialKappa Target m ((insert v ({none}:Finset (Option (Fin (t+1))))).image q)
    ((((d+1:ℕ):ℝ)+2)/((L₀+1:ℕ):ℝ))*(∫ z, scheduledShapeEnergy (L₀+1) (d+1) (.node k hk C) (rootSchedule (r+1+(d+1)) (fun j v => (Q (e j v)).val)) (K z) (f z) ∂μ) ≤
      2*(2*((L₀+1:ℕ):ℝ)⁻¹ + |physicalTreeMatrixCovariance (gridExponents L₀) S x M θB hB N u Target q|/|J|+
        (2*Real.sqrt W'+(∫ z, h z ∂μ))*shiftedCharge B Target S (v::cs).length/|J|+
        pairHistoryMass S m x*(2*Real.sqrt W))+
      ((((d+1:ℕ):ℝ)+2)/((L₀+1:ℕ):ℝ))*(16*W)

Definitions. The statement is written with OAI.DilutedSpinGlass.FinitePath, OAI.DilutedSpinGlass.FullRootState, OAI.DilutedSpinGlass.KernelTower, OAI.DilutedSpinGlass.PrescribedTree, OAI.DilutedSpinGlass.ReducedTopology, OAI.DilutedSpinGlass.RootPath, OAI.DilutedSpinGlass.fullRootLaw, OAI.DilutedSpinGlass.kernelHeightCast, OAI.DilutedSpinGlass.rootArray, OAI.DilutedSpinGlass.rootPathMeasurable, OAI.DilutedSpinGlass.scoreRate, OAI.DilutedSpinGlass.vectorHeightCast, OAI.DilutedSpinGlass.ConcreteReservoir.Bond, OAI.DilutedSpinGlass.ConcreteReservoir.Site, OAI.DilutedSpinGlass.ConcreteReservoir.bondLaw, OAI.DilutedSpinGlass.ConcreteReservoir.markLaw, OAI.DilutedSpinGlass.ConcreteReservoir.observableAt, OAI.DilutedSpinGlass.ConcreteReservoir.physicalBase, OAI.DilutedSpinGlass.ConcreteReservoir.readSpin, OAI.DilutedSpinGlass.DepthAverage.Regular, OAI.DilutedSpinGlass.FiniteLaw.uniform, OAI.DilutedSpinGlass.HeterogeneousMarks.Row, OAI.DilutedSpinGlass.HeterogeneousMarks.dictionaryFactor, OAI.DilutedSpinGlass.HeterogeneousMarks.rootAlphabet, OAI.DilutedSpinGlass.HeterogeneousMarks.rootTower, OAI.DilutedSpinGlass.HeterogeneousMarks.rootVector, OAI.DilutedSpinGlass.KernelTower.terminalState, OAI.DilutedSpinGlass.KernelTower.terminalTower, OAI.DilutedSpinGlass.PrescribedTree.Leaf, OAI.DilutedSpinGlass.PrescribedTree.canonicalMatrixTail, OAI.DilutedSpinGlass.PrescribedTree.heightCast, OAI.DilutedSpinGlass.PrescribedTree.leafDecidableEq, OAI.DilutedSpinGlass.PrescribedTree.leafHeightCast, OAI.DilutedSpinGlass.PrescribedTree.pairHistoryMass, OAI.DilutedSpinGlass.PrescribedTree.partialKappa, OAI.DilutedSpinGlass.PrescribedTree.shiftedCharge, OAI.DilutedSpinGlass.PrescribedTree.shiftedPrefixDepths, OAI.DilutedSpinGlass.PrescribedTree.splitFrame, OAI.DilutedSpinGlass.PrescribedTree.splitFrameLeaf, OAI.DilutedSpinGlass.PrescribedTree.stem, OAI.DilutedSpinGlass.ReducedTopology.Admissible, OAI.DilutedSpinGlass.ReducedTopology.Vertex, OAI.DilutedSpinGlass.ReducedTopology.projectionShiftError, OAI.DilutedSpinGlass.ReducedTopology.realize, OAI.DilutedSpinGlass.ReducedTopology.rootSchedule, OAI.DilutedSpinGlass.ReducedTopology.scheduledShapeEnergy, OAI.DilutedSpinGlass.ReducedTopology.shiftedFrameHeight, OAI.DilutedSpinGlass.UniversalDictionary.Alphabet, OAI.DilutedSpinGlass.UniversalDictionary.Labels, OAI.DilutedSpinGlass.UniversalDictionary.Spec, OAI.DilutedSpinGlass.UniversalDictionary.Table, OAI.DilutedSpinGlass.UniversalDictionary.anchor, OAI.DilutedSpinGlass.UniversalDictionary.direction, OAI.DilutedSpinGlass.UniversalDictionary.gridExponents, OAI.DilutedSpinGlass.UniversalDictionary.physicalTreeMatrixCovariance, OAI.DilutedSpinGlass.UniversalDictionary.prior, OAI.DilutedSpinGlass.UniversalDictionary.readVector, OAI.DilutedSpinGlass.UniversalDictionary.specMeasurable and OAI.DilutedSpinGlass.UniversalDictionary.weights (definitions bundle OAIQ041D000 of this split); OAI.DilutedSpinGlass.Interaction, OAI.DilutedSpinGlass.InteractionSample, OAI.DilutedSpinGlass.Model and OAI.DilutedSpinGlass.Spin (the published bundle Def_DilutedSpin).

Used by. The proofs of OAI_DilutedSpinGlass_UniversalDictionary_physicalOverlapLimsup_tendsto and OAI.DilutedSpinGlass.UniversalDictionary.physicalShapeLimsup_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
section
namespace DilutedSpinGlass.UniversalDictionary
open _root_.MeasureTheory ProbabilityTheory HeterogeneousMarks PhysicalRoot PrescribedTree ConcreteReservoir
open ReducedTopology
open scoped NNReal BigOperators
variable {α : Type} [Fintype α] [DecidableEq α] {p t : ℕ}
attribute [local irreducible] projectionShiftError matrixProjectionError oldProjectionError
    matrixObservableHistory KernelTower.halfTripleDifferenceAt splitProjector

theorem physical_grid_scheduled_step (M : Model p) (θB hB : ℝ) (N H r d : ℕ)
    (u : Spec (H+1+1+r+1+d) × ℕ → ℝ)
    (k : ℕ+) (hk : 2≤(k:ℕ)) (a : α) (C : Fin k → ReducedTopology)
    (e : (j : Fin k) → (C j).Vertex → {j : α // j≠a})
    (Q : α → Fin (H+1+1+r+1+(d+1))) (ha : (Q a).val=r+1+(d+1))
    (had : ∀ j, ReducedTopology.Admissible (C j) (fun v => (Q (e j v)).val) (r+1+(d+1)+1) (r+1+(d+1)+1+H))
    {η : ℝ} (hlarge : 1<η*(H+1+1+r+1+(d+1):ℕ)) (hr : DepthAverage.Regular η Q) :
    let L₀ := H+1+1+r+1+d
    let μ := fullRootLaw (fun _ : Fin N => M.field.toMeasure) (bondLaw M N)
      (markLaw (weights L₀) N) (M.alpha*N) (scoreRate N)
    let K := rootTower (KernelTower.terminalTower (fun _ : Fin N => false) FiniteLaw.uniform L₀)
      (fun i : Labels L₀ (Site N) => prior i.1.1) (gridExponents L₀) (physicalBase M θB hB N)
      (dictionaryFactor (observableAt direction N) (observableAt anchor N) u)
    let f := rootVector (readVector (fun v x => readSpin (KernelTower.terminalState L₀ x) v))
      (I := Labels L₀ (Site N)) (A := fun i => Alphabet i.1.1) (X := Bond p N) (Y := ℝ) (M := N)
    let ht := shiftedFrameHeight H r (d+1)
    let OldChild := fun j => realize (H+1) (r+1+(d+1)+1) (C j) (fun v => (Q (e j v)).val)
    let NewChild := fun j => realize H (r+1+1+(d+1)+1) (C j) (fun v => (Q (e j v)).val+1)
    let OldNode := PrescribedTree.node k OldChild
    let NewNode := PrescribedTree.node k NewChild
    let BaseNode := PrescribedTree.node k (fun j => realize H (r+1+(d+1)+1) (C j) (fun v => (Q (e j v)).val))
    let B := shiftedPrefixDepths (stem BaseNode r) (d+1)
    let S := splitFrame OldNode r (d+1)
    let RawTarget := splitFrame NewNode (r+1) (d+1)
    let Target := heightCast ht RawTarget
    let _ := fun z => kernelHeightCast ht.symm (K z)
    let _ := fun z => vectorHeightCast ht.symm (f z)
    let W := (k:ℝ)*∑ j, Real.sqrt (∫ z, scheduledShapeEnergy (L₀+1) (r+1+(d+1)) (C j) (fun v => (Q (e j v)).val) (K z) (f z) ∂μ)
    let W' := (k:ℝ)*∑ j, Real.sqrt (∫ z, scheduledShapeEnergy (L₀+1) (r+1+1+(d+1)) (C j) (fun v => (Q (e j v)).val+1) (K z) (f z) ∂μ)
    let h := fun z => projectionShiftError a C e (K z) (f z) Q
    ∀ (b : OldNode.Leaf) (b' : NewNode.Leaf) (q0 : Option (Fin (t+1)) → RawTarget.Leaf),
      Function.Bijective q0 → q0 none=splitFrameLeaf NewNode (r+1) (d+1) 0 b' →
        q0 (some (Fin.last t))=splitFrameLeaf NewNode (r+1) (d+1) 1 b' →
    let v := some (Fin.last t)
    let cs := canonicalMatrixTail t
    let q := fun j => leafHeightCast ht RawTarget (q0 j)
    let x := splitFrameLeaf OldNode r (d+1) 0 b
    let m := Fin.cons 0 (gridExponents L₀)
    let J := partialKappa Target m (Finset.univ.image q)/
      partialKappa Target m ((insert v ({none}:Finset (Option (Fin (t+1))))).image q)
    ((((d+1:ℕ):ℝ)+2)/((L₀+1:ℕ):ℝ))*(∫ z, scheduledShapeEnergy (L₀+1) (d+1) (.node k hk C) (rootSchedule (r+1+(d+1)) (fun j v => (Q (e j v)).val)) (K z) (f z) ∂μ) ≤
      2*(2*((L₀+1:ℕ):ℝ)⁻¹ + |physicalTreeMatrixCovariance (gridExponents L₀) S x M θB hB N u Target q|/|J|+
        (2*Real.sqrt W'+(∫ z, h z ∂μ))*shiftedCharge B Target S (v::cs).length/|J|+
        pairHistoryMass S m x*(2*Real.sqrt W))+
      ((((d+1:ℕ):ℝ)+2)/((L₀+1:ℕ):ℝ))*(16*W) := by
  sorry

end DilutedSpinGlass.UniversalDictionary
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/PhysicalScheduledStep.lean (declaration OAI.DilutedSpinGlass.UniversalDictionary.physical_grid_scheduled_step)

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