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

Proved

OAI.SKGapCutoff.Recipe.literal_local_weak_comparison_targets

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.SKGapCutoff.Recipe.literal_local_weak_comparison_targets of lean/OAI/Probability/SKGap/Localization/LocalFixedRecipe.lean at commit 1581d4c, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.SKGapCutoff.Recipe, with open scoped BigOperators Matrix.Norms.Frobenius, open Primary Static SKGap.Stein SKGap.Noncrossing.Primary SKGap.Noncrossing.Primary.Tensor.Series SKGap.Noncrossing.ClosedMarked):

theorem literal_local_weak_comparison_targets (j : ℝ) (J : ∀b,Interaction (n b))
    (h : ∀b,Fin (n b)→ℝ) (l : Fin M) (f : KernelExpr)
    (a c : ∀b,Spin (n b)→ℝ) (r e : ∀b,Fin (n b)→ℝ)
    (w y : ∀b,VectorFields (n b)) (E : ∀b,Set (Spin (n b)))
    (hn : ∀b,0<n b) (he : ∀b,vectorNorm (e b)≤1)
    {K B R Bc Va Vc V S F W C : ℝ}
    (hK : 0≤K) (hB : 0≤B) (hBc : 0≤Bc) (hVa : 0≤Va) (hVc : 0≤Vc)
    (hV : 0≤V) (hS : 0≤S) (hF : 0≤F) (hW : 0≤W) (hC : 0≤C)
    (hop : ∀b,SKGap.opNorm (J b)≤K)
    (hformal : ∀b x,∀q<M,ShapeBound (formalField j (J b) (h b) x q) B)
    (ha : ∀b x,x∈flipNeighborhood (flipNeighborhood (E b))→|a b x|≤R)
    (hc : ∀b x,x∈flipNeighborhood (flipNeighborhood (E b))→|Real.sqrt (n b:ℝ)*c b x|≤Bc)
    (hDa : ∀b x,x∈flipNeighborhood (E b)→‖derivativeVector (a b) x‖≤Va)
    (hDc : ∀b x,x∈flipNeighborhood (E b)→‖derivativeVector (fun v=>Real.sqrt (n b:ℝ)*c b v) x‖≤Vc)
    (heq : ∀b x,x∈E b→LiteralEquations (J b) j f (fld j (J b) (h b) l.val)
      (mag j (J b) (h b) l.val) (w b) (y b) (a b) (c b) (r b) (e b) x)
    (hdiag : ∀b x,x∈flipNeighborhood (E b)→LiteralInitialDiagnostics
      (literalInitial (J b) j (Real.sqrt (n b:ℝ)) f (fld j (J b) (h b) l.val)
        (mag j (J b) (h b) l.val) (a b) (c b) (r b) (e b))
      (w b) (y b) (primaryTree j (J b) (h b) x l.val) x V S F (literalResidualBudget j f R Bc l.val))
    (P : ∀b,Observables (n b)) (hP : ∀b x,0≤P b x) (hp : ∀b,∑x,P b x=1)
    (hm : ∀b x i,conditionalMean (P b) x i=mag j (J b) (h b) 1 x i)
    (hJ : ∀b,(J b).IsSymm)
    (hw : ∀b x,x∈flipNeighborhood (E b)→ClosedWordTestBound j
      (fun i=>phi (fld j (J b) (h b) l.val x i) (r b i) (a b x)) (J b) (literalResidualBudget j f R Bc l.val) W (2*(2+2*l.val)+5))
    (hd : ∀b x,x∈flipNeighborhood (E b)→ClosedWordDiagramBound j
      (fun i=>phi (fld j (J b) (h b) l.val x i) (r b i) (a b x)) (J b) (literalResidualBudget j f R Bc l.val) C (2*M+2*(2+2*l.val)+4)) :
    ((∀p≤l.val,LocalUniformWeak E P (fun b x=>∑i,residual j (J b) (h b) p x i*w b x i)) ∧
    LocalUniformWeak E P (fun b x=>∑i,residual j (J b) (h b) l.val x i*y b x i) ∧
    LocalUniformWeak E P (fun b x=>∑i,residual j (J b) (h b) l.val x i*
      (Real.tanh (fld j (J b) (h b) l.val x i)/Real.sqrt (n b:ℝ))) ∧
    LocalUniformMultiplier E (fun b=>siteMean (fun x i=>phi (fld j (J b) (h b) l.val x i) (r b i) (a b x))) ∧
    LocalUniformMultiplier E (fun b x=>j*onsager j (J b) (h b) l.val x) ∧
    LocalUniformMultiplier E (fun b x=>Real.sqrt (n b:ℝ)*c b x)) ∧
    LocalUniformWeak E P (fun b x=>j*c b x*(∑i,residual j (J b) (h b) l.val x i*
      (Real.tanh (fld j (J b) (h b) l.val x i)-Real.tanh (r b i))))

Definitions. The statement is written with OAI.SKGap.opNorm, OAI.SKGapCutoff.Interaction, OAI.SKGapCutoff.Observables, OAI.SKGapCutoff.Spin, OAI.SKGapCutoff.VectorFields, OAI.SKGap.Stein.KernelExpr, OAI.SKGap.Stein.phi, OAI.SKGapCutoff.Primary.ShapeBound, OAI.SKGapCutoff.Primary.derivativeVector, OAI.SKGapCutoff.Primary.fld, OAI.SKGapCutoff.Primary.formalField, OAI.SKGapCutoff.Primary.mag, OAI.SKGapCutoff.Primary.onsager, OAI.SKGapCutoff.Primary.primaryTree, OAI.SKGapCutoff.Primary.residual, OAI.SKGapCutoff.Primary.siteMean, OAI.SKGapCutoff.Recipe.ClosedWordDiagramBound, OAI.SKGapCutoff.Recipe.ClosedWordTestBound, OAI.SKGapCutoff.Recipe.vectorNorm, OAI.SKGapCutoff.Static.LocalUniformMultiplier, OAI.SKGapCutoff.Static.conditionalMean and OAI.SKGapCutoff.Static.flipNeighborhood (definitions bundle OAIQ072A000 of this split); OAI.SKGapCutoff.Recipe.LiteralEquations, OAI.SKGapCutoff.Recipe.LiteralInitialDiagnostics, OAI.SKGapCutoff.Recipe.literalInitial, OAI.SKGapCutoff.Recipe.literalResidualBudget and OAI.SKGapCutoff.Static.LocalUniformWeak (definitions bundle OAIQ072A002 of this split).

Used by. The proof of OAI.SKGapCutoff.Recipe.buffered_root_source_weak of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIQ072A002

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

noncomputable section
open scoped BigOperators Matrix.Norms.Frobenius
namespace SKGapCutoff.Recipe
open Primary Static SKGap.Stein SKGap.Noncrossing.Primary SKGap.Noncrossing.Primary.Tensor.Series SKGap.Noncrossing.ClosedMarked
universe u
variable {Ω : Type u} {n : Ω→ℕ} {M : ℕ}
theorem literal_local_weak_comparison_targets (j : ℝ) (J : ∀b,Interaction (n b))
    (h : ∀b,Fin (n b)→ℝ) (l : Fin M) (f : KernelExpr)
    (a c : ∀b,Spin (n b)→ℝ) (r e : ∀b,Fin (n b)→ℝ)
    (w y : ∀b,VectorFields (n b)) (E : ∀b,Set (Spin (n b)))
    (hn : ∀b,0<n b) (he : ∀b,vectorNorm (e b)≤1)
    {K B R Bc Va Vc V S F W C : ℝ}
    (hK : 0≤K) (hB : 0≤B) (hBc : 0≤Bc) (hVa : 0≤Va) (hVc : 0≤Vc)
    (hV : 0≤V) (hS : 0≤S) (hF : 0≤F) (hW : 0≤W) (hC : 0≤C)
    (hop : ∀b,SKGap.opNorm (J b)≤K)
    (hformal : ∀b x,∀q<M,ShapeBound (formalField j (J b) (h b) x q) B)
    (ha : ∀b x,x∈flipNeighborhood (flipNeighborhood (E b))→|a b x|≤R)
    (hc : ∀b x,x∈flipNeighborhood (flipNeighborhood (E b))→|Real.sqrt (n b:ℝ)*c b x|≤Bc)
    (hDa : ∀b x,x∈flipNeighborhood (E b)→‖derivativeVector (a b) x‖≤Va)
    (hDc : ∀b x,x∈flipNeighborhood (E b)→‖derivativeVector (fun v=>Real.sqrt (n b:ℝ)*c b v) x‖≤Vc)
    (heq : ∀b x,x∈E b→LiteralEquations (J b) j f (fld j (J b) (h b) l.val)
      (mag j (J b) (h b) l.val) (w b) (y b) (a b) (c b) (r b) (e b) x)
    (hdiag : ∀b x,x∈flipNeighborhood (E b)→LiteralInitialDiagnostics
      (literalInitial (J b) j (Real.sqrt (n b:ℝ)) f (fld j (J b) (h b) l.val)
        (mag j (J b) (h b) l.val) (a b) (c b) (r b) (e b))
      (w b) (y b) (primaryTree j (J b) (h b) x l.val) x V S F (literalResidualBudget j f R Bc l.val))
    (P : ∀b,Observables (n b)) (hP : ∀b x,0≤P b x) (hp : ∀b,∑x,P b x=1)
    (hm : ∀b x i,conditionalMean (P b) x i=mag j (J b) (h b) 1 x i)
    (hJ : ∀b,(J b).IsSymm)
    (hw : ∀b x,x∈flipNeighborhood (E b)→ClosedWordTestBound j
      (fun i=>phi (fld j (J b) (h b) l.val x i) (r b i) (a b x)) (J b) (literalResidualBudget j f R Bc l.val) W (2*(2+2*l.val)+5))
    (hd : ∀b x,x∈flipNeighborhood (E b)→ClosedWordDiagramBound j
      (fun i=>phi (fld j (J b) (h b) l.val x i) (r b i) (a b x)) (J b) (literalResidualBudget j f R Bc l.val) C (2*M+2*(2+2*l.val)+4)) :
    ((∀p≤l.val,LocalUniformWeak E P (fun b x=>∑i,residual j (J b) (h b) p x i*w b x i)) ∧
    LocalUniformWeak E P (fun b x=>∑i,residual j (J b) (h b) l.val x i*y b x i) ∧
    LocalUniformWeak E P (fun b x=>∑i,residual j (J b) (h b) l.val x i*
      (Real.tanh (fld j (J b) (h b) l.val x i)/Real.sqrt (n b:ℝ))) ∧
    LocalUniformMultiplier E (fun b=>siteMean (fun x i=>phi (fld j (J b) (h b) l.val x i) (r b i) (a b x))) ∧
    LocalUniformMultiplier E (fun b x=>j*onsager j (J b) (h b) l.val x) ∧
    LocalUniformMultiplier E (fun b x=>Real.sqrt (n b:ℝ)*c b x)) ∧
    LocalUniformWeak E P (fun b x=>j*c b x*(∑i,residual j (J b) (h b) l.val x i*
      (Real.tanh (fld j (J b) (h b) l.val x i)-Real.tanh (r b i)))) := by
  sorry

end SKGapCutoff.Recipe

end
end

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/Localization/LocalFixedRecipe.lean (declaration OAI.SKGapCutoff.Recipe.literal_local_weak_comparison_targets)

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