OAI.SharpRamseyFive.ScoreGeometry.eventually_four_low_protocol: intermediate lemma from OpenAI, Sharp logarithmic exponents for fixed off-diagonal Ramsey numbers (result family 170)

Proved

OAI.SharpRamseyFive.ScoreGeometry.eventually_four_low_protocol

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.SharpRamseyFive.main, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, Sharp logarithmic exponents for fixed off-diagonal Ramsey numbers (result family 170), OpenAI Math Release, September 24, 2026, result family 170, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.SharpRamseyFive.ScoreGeometry.eventually_four_low_protocol of lean/OAI/Combinatorics/RamseyFive/Geometry/FiniteFamily.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.SharpRamseyFive.ScoreGeometry, with open Module ProjectiveIncidence ProjectiveTraining GreedyTraining GlobalRadial, open CellVariance ScoreRegularity PoissonScore WeightedPrograms MeasureTheory, open Filter ParameterHierarchy MeasurePublicTable Metadata FiniteEntropy, open scoped BigOperators LinearAlgebra.Projectivization Classical NNReal Topology):

theorem eventually_four_low_protocol {η : ℝ} (hη : 0<η) (hη' : η<1/10)
    (Cb : ℝ) (hCb : 0≤Cb) :
    ∀ᶠ σ : ℝ in atTop,∀ (D b τ g : ℝ) (R : ℕ) (L₀ : ℝ≥0),
    ∀ (q : ℕ) (K I J : Type) [Field K] [Finite K] [CharP K q] [Fintype I] [LinearOrder J]
      [Fintype (I→K)] [Fintype (ℙ K (I→K))] [Fintype (ℙ K (Dual K (I→K)))]
      [∀x : ℙ K (I→K),Fintype (RadialLine x)],
    ∀ (F : Finset J) (hF : F.Nonempty) (Flat : J→Submodule K (I→K))
      (X U : Finset (ℙ K (I→K))) (T : Finset (ℙ K (Dual K (I→K)))),
      Nat.card K=q → Real.exp σ=q → Fintype.card I=5 →
      Range η σ D R → (L₀:ℝ)=L η σ D → 0≤b → b≤Cb*D*σ^(6*beta η) →
      0<τ → τ≤σ^(-200*beta η) → X⊆U → X.card≤T.card →
      (Nat.card K:ℝ)*(incidences X T:ℝ)≤τ*X.card*T.card →
      (Nat.card K:ℝ)^5*Real.exp (-b)≤(X.card:ℝ)*T.card →
      (X.card:ℝ)=Real.exp (3*σ/2+g) →
      100*(Nat.card K:ℝ)*P η σ D R<(X.card:ℝ) →
      (X.card:ℝ)≤(Nat.card K:ℝ)^2*Real.exp (P η σ D R/10000) →
      (∀j∈F,finrank K (Flat j)=3) →
      (∀V : Submodule K (I→K),finrank K V=3 → ∃j∈F,Flat j=V) →
    let t := (Real.exp (3*σ/2+g))^(4/3:ℝ)/Real.exp σ*Real.exp (-g/5)
    let ht : 0<t := by positivity
    let S := peelSet (P η σ D R/10000<g) F hF (fun j=>flatPoints (Flat j)) X ⌈t⌉₊ (Nat.ceil_pos.mpr ht)
    let m := peelLength (P η σ D R/10000<g) F hF (fun j=>flatPoints (Flat j)) X ⌈t⌉₊ (Nat.ceil_pos.mpr ht)
    let C := clippedPart S F hF (fun j=>flatPoints (Flat j)) X m
    (∀i,(C i).card≤(S.card:ℝ)/25) →
    let N := scoreCutoff S U (P η σ D R) τ
    Real.log (Nat.card (TrainingCode (I→K) (listCap σ) (productCap σ) (Nat.card (I→K))))≤q ∧
    Real.log N≤Real.log (2*(Nat.card K:ℝ))+scoreSearchCost S U (P η σ D R) τ ∧
    Real.log (Fintype.card (ℙ K (I→K))+1:ℝ)+
      Real.log (Fintype.card (ScoredPayload U S.card σ (P η σ D R) τ):ℝ)≤
        100*(Nat.card K:ℝ)*P η σ D R*(Real.log ((U.card:ℝ)/X.card)+P η σ D R) ∧
    ∃c : TrainingCode (I→K) (listCap σ) (productCap σ) (Nat.card (I→K)),
    let n : Fin (Fintype.card (ℙ K (I→K))+1) := ⟨S.card,Nat.lt_succ_of_le (Finset.card_le_univ S)⟩
    let p₀ := finiteImageLaw (baseTapeMeasure U (P η σ D R) τ R L₀)
      (fun t=>(scoredEncoded X U n σ (P η σ D R) τ (9/20) 10 R L₀ c t).map
        (scoredMessageDecoded U σ (P η σ D R) τ R L₀ t))
    p₀ none≤Real.exp (-(Nat.card K:ℝ)) ∧
    (∀W,0<p₀ (some W)→W⊆U ∧ (W.card:ℝ)≤(X.card:ℝ)*Real.exp (10*P η σ D R) ∧
      (9/20:ℝ)*X.card≤(W∩X).card)

Definitions. The statement is written with OAI.SharpRamseyFive.FiniteEntropy.Law, OAI.SharpRamseyFive.FiniteEntropy.finiteImageLaw and OAI.SharpRamseyFive.ProjectiveIncidence.RadialLine (definitions bundle OAIQ060R000 of this split); OAI.SharpRamseyFive.GreedyTraining.clippedPart, OAI.SharpRamseyFive.GreedyTraining.peelLength, OAI.SharpRamseyFive.GreedyTraining.peelSet, OAI.SharpRamseyFive.Metadata.TrainingCode, OAI.SharpRamseyFive.Metadata.instFintypeTrainingCode, OAI.SharpRamseyFive.Metadata.listCap, OAI.SharpRamseyFive.Metadata.productCap, OAI.SharpRamseyFive.ParameterHierarchy.L, OAI.SharpRamseyFive.ParameterHierarchy.P, OAI.SharpRamseyFive.ParameterHierarchy.Range, OAI.SharpRamseyFive.ParameterHierarchy.beta, OAI.SharpRamseyFive.ProjectiveTraining.flatPoints, OAI.SharpRamseyFive.ScoreGeometry.BaseTape, OAI.SharpRamseyFive.ScoreGeometry.ScoredMessage, OAI.SharpRamseyFive.ScoreGeometry.ScoredPayload, OAI.SharpRamseyFive.ScoreGeometry.scoreCutoff, OAI.SharpRamseyFive.ScoreGeometry.scoreSearchCost and OAI.SharpRamseyFive.ScoreGeometry.sizeScoreCutoff (definitions bundle OAIQ060R001 of this split); OAI.SharpRamseyFive.ProjectiveIncidence.incidences (definitions bundle OAIQ060R002 of this split); OAI.SharpRamseyFive.ScoreGeometry.baseTapeMeasure, OAI.SharpRamseyFive.ScoreGeometry.instIsProbabilityMeasureBaseTapeBaseTapeMeasure, OAI.SharpRamseyFive.ScoreGeometry.scoredEncoded and OAI.SharpRamseyFive.ScoreGeometry.scoredMessageDecoded (definitions bundle OAIQ060R003 of this split).

Used by. The proof of OAI.SharpRamseyFive.ScoreGeometry.eventually_four_finite_predictor of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIQ060R002
import Definitions.Def_OAIQ060R003

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

namespace SharpRamseyFive.ScoreGeometry

open Module ProjectiveIncidence ProjectiveTraining GreedyTraining GlobalRadial
open CellVariance ScoreRegularity PoissonScore WeightedPrograms MeasureTheory
open Filter ParameterHierarchy MeasurePublicTable Metadata FiniteEntropy
open scoped BigOperators LinearAlgebra.Projectivization Classical NNReal Topology

theorem eventually_four_low_protocol {η : ℝ} (hη : 0<η) (hη' : η<1/10)
    (Cb : ℝ) (hCb : 0≤Cb) :
    ∀ᶠ σ : ℝ in atTop,∀ (D b τ g : ℝ) (R : ℕ) (L₀ : ℝ≥0),
    ∀ (q : ℕ) (K I J : Type) [Field K] [Finite K] [CharP K q] [Fintype I] [LinearOrder J]
      [Fintype (I→K)] [Fintype (ℙ K (I→K))] [Fintype (ℙ K (Dual K (I→K)))]
      [∀x : ℙ K (I→K),Fintype (RadialLine x)],
    ∀ (F : Finset J) (hF : F.Nonempty) (Flat : J→Submodule K (I→K))
      (X U : Finset (ℙ K (I→K))) (T : Finset (ℙ K (Dual K (I→K)))),
      Nat.card K=q → Real.exp σ=q → Fintype.card I=5 →
      Range η σ D R → (L₀:ℝ)=L η σ D → 0≤b → b≤Cb*D*σ^(6*beta η) →
      0<τ → τ≤σ^(-200*beta η) → X⊆U → X.card≤T.card →
      (Nat.card K:ℝ)*(incidences X T:ℝ)≤τ*X.card*T.card →
      (Nat.card K:ℝ)^5*Real.exp (-b)≤(X.card:ℝ)*T.card →
      (X.card:ℝ)=Real.exp (3*σ/2+g) →
      100*(Nat.card K:ℝ)*P η σ D R<(X.card:ℝ) →
      (X.card:ℝ)≤(Nat.card K:ℝ)^2*Real.exp (P η σ D R/10000) →
      (∀j∈F,finrank K (Flat j)=3) →
      (∀V : Submodule K (I→K),finrank K V=3 → ∃j∈F,Flat j=V) →
    let t := (Real.exp (3*σ/2+g))^(4/3:ℝ)/Real.exp σ*Real.exp (-g/5)
    let ht : 0<t := by positivity
    let S := peelSet (P η σ D R/10000<g) F hF (fun j=>flatPoints (Flat j)) X ⌈t⌉₊ (Nat.ceil_pos.mpr ht)
    let m := peelLength (P η σ D R/10000<g) F hF (fun j=>flatPoints (Flat j)) X ⌈t⌉₊ (Nat.ceil_pos.mpr ht)
    let C := clippedPart S F hF (fun j=>flatPoints (Flat j)) X m
    (∀i,(C i).card≤(S.card:ℝ)/25) →
    let N := scoreCutoff S U (P η σ D R) τ
    Real.log (Nat.card (TrainingCode (I→K) (listCap σ) (productCap σ) (Nat.card (I→K))))≤q ∧
    Real.log N≤Real.log (2*(Nat.card K:ℝ))+scoreSearchCost S U (P η σ D R) τ ∧
    Real.log (Fintype.card (ℙ K (I→K))+1:ℝ)+
      Real.log (Fintype.card (ScoredPayload U S.card σ (P η σ D R) τ):ℝ)≤
        100*(Nat.card K:ℝ)*P η σ D R*(Real.log ((U.card:ℝ)/X.card)+P η σ D R) ∧
    ∃c : TrainingCode (I→K) (listCap σ) (productCap σ) (Nat.card (I→K)),
    let n : Fin (Fintype.card (ℙ K (I→K))+1) := ⟨S.card,Nat.lt_succ_of_le (Finset.card_le_univ S)⟩
    let p₀ := finiteImageLaw (baseTapeMeasure U (P η σ D R) τ R L₀)
      (fun t=>(scoredEncoded X U n σ (P η σ D R) τ (9/20) 10 R L₀ c t).map
        (scoredMessageDecoded U σ (P η σ D R) τ R L₀ t))
    p₀ none≤Real.exp (-(Nat.card K:ℝ)) ∧
    (∀W,0<p₀ (some W)→W⊆U ∧ (W.card:ℝ)≤(X.card:ℝ)*Real.exp (10*P η σ D R) ∧
      (9/20:ℝ)*X.card≤(W∩X).card) := by
  sorry

end SharpRamseyFive.ScoreGeometry

end OAI
end

Source

OpenAI, Sharp logarithmic exponents for fixed off-diagonal Ramsey numbers (result family 170), OpenAI Math Release, September 24, 2026, https://github.com/openai/math (result family 170; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Sharp-Logarithmic-Exponents-for-Fixed-Off-Diagonal-Ramsey-Numbers-September-24-2026/paper.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/RamseyFive/Geometry/FiniteFamily.lean (declaration OAI.SharpRamseyFive.ScoreGeometry.eventually_four_low_protocol)

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