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

Proved

OAI.SharpRamseyFive.ScoreGeometry.cell_score_from_integrals

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.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.cell_score_from_integrals of lean/OAI/Combinatorics/RamseyFive/Iteration/CellScoreProcedure.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 CellVariance ScoreRegularity, open MeasureTheory ProbabilityTheory PoissonScore, open scoped BigOperators LinearAlgebra.Projectivization Classical NNReal):

theorem cell_score_from_integrals {d : ℕ} (hdim : finrank K V=d+1) (hd : 1≤d)
    (hq : (10:ℝ)≤Nat.card K) (X U S : Finset (ℙ K V)) (hSX : S⊆X) (hSU : S⊆U)
    (hS : S.Nonempty) (hret : X.card≤4*S.card)
    {J : Type*} [Fintype J] (C : J→Finset (ℙ K V))
    (hC : (∑j,(C j).card)≤3*S.card) (ia ib : ℙ K V→J)
    (hf : ∀x,ownFraction S (C (ia x)) (C (ib x))≤2/25)
    (T : Finset (ℙ K (Dual K V))) (L : ℝ≥0) (R : ℕ) (b τ CP : ℝ)
    (hτ : 0<τ) (hτhi : 8*τ≤4/5)
    (hdensity : (Nat.card K:ℝ)*(incidences X T:ℝ)≤τ*X.card*T.card)
    (hprod : (Nat.card K:ℝ)^(d+1)*Real.exp (-b)≤(X.card:ℝ)*T.card)
    (hP : 104≤(L:ℝ)*R) (hCP : 0≤CP)
    (hnP : (Nat.card K:ℝ)*((L:ℝ)*R)≤S.card)
    (hbad : ((irregular (d:=d) S C ((L:ℝ)/100)).card:ℝ)≤(S.card:ℝ)/100)
    (he : b+16*((L:ℝ)*R)*τ+Real.log 16+Real.log 3200000000≤((L:ℝ)*R)/2000)
    (hs : 8*((L:ℝ)*R)*τ+Real.log 1600≤(L:ℝ)/1000)
    (hu : 8*((L:ℝ)*R)*τ+Real.log 16≤(L:ℝ)*R)
    (hk : b+8*((L:ℝ)*R)*τ+Real.log 16≤(L:ℝ)*R)
    (hκL : b+8*((L:ℝ)*R)*τ+Real.log 16+Real.log 100≤(91/100:ℝ)*(L:ℝ))
    (hR : (R:ℝ)*Real.exp (-(23/25:ℝ)*(L:ℝ))≤1/10)
    (htrain : (∫ω,((typicalFailures S S (fun x => C (ia x)∪C (ib x)) (exceptional S C)
      (fun x => Real.exp (-(L:ℝ)*(1-ownFraction S (C (ia x)) (C (ib x)))))
      (scale (K:=K) d S.card*Real.exp (-(b+8*((L:ℝ)*R)*τ+Real.log 16))/10) ω).card:ℝ)
      ∂scheduleMeasure (fun _ : S => L*pointStrength S) R)≤(S.card:ℝ)*Real.exp (-(L:ℝ)/1000))
    (hamb : (∫ω,((typicalFailures U S (fun x => C (ia x)∪C (ib x)) (exceptional S C)
      (fun x => Real.exp (-(L:ℝ)*(1-ownFraction S (C (ia x)) (C (ib x)))))
      (scale (K:=K) d S.card*Real.exp (-(b+8*((L:ℝ)*R)*τ+Real.log 16))/10) ω).card:ℝ)
      ∂scheduleMeasure (fun _ : S => L*pointStrength S) R)≤(S.card:ℝ)*Real.exp (CP*((L:ℝ)*R))) :
    Real.exp (-8*((L:ℝ)*R)*τ)/4≤
    (scheduleMeasure (fun _ : S => L*pointStrength S) R).real
      {ω | let W := decoded U S (fun x => C (ia x)∪C (ib x)) (fun x => Real.exp (-(L:ℝ)*(1-ownFraction S (C (ia x)) (C (ib x))))) (emptyTests (exceptional S C) (hyperplaneSupport S) ω).card ω
        (sampleCount ω:ℝ)≤2*((Nat.card K:ℝ)*((L:ℝ)*R)) ∧
        (W.card:ℝ)≤(S.card:ℝ)*Real.exp ((CP+2)*((L:ℝ)*R)) ∧
        (9/10:ℝ)*S.card≤((W∩S).card:ℝ)}

Definitions. The statement is written with OAI.SharpRamseyFive.PoissonScore.sampleCount, OAI.SharpRamseyFive.ScoreGeometry.pointStrength, OAI.SharpRamseyFive.ScoreRegularity.ownFraction and OAI.SharpRamseyFive.ScoreRegularity.scale (definitions bundle OAIQ060R001 of this split); OAI.SharpRamseyFive.ProjectiveIncidence.incidences (definitions bundle OAIQ060R002 of this split); OAI.SharpRamseyFive.PoissonScore.emptyTests, OAI.SharpRamseyFive.PoissonScore.scheduleMeasure, OAI.SharpRamseyFive.ScoreGeometry.decoded, OAI.SharpRamseyFive.ScoreGeometry.hyperplaneSupport, OAI.SharpRamseyFive.ScoreGeometry.typicalFailures and OAI.SharpRamseyFive.ScoreRegularity.exceptional (definitions bundle OAIQ060R003 of this split); OAI.SharpRamseyFive.ScoreRegularity.irregular (definitions bundle OAIQ060R004 of this split).

Used by. The proofs of OAI.SharpRamseyFive.ScoreGeometry.eventually_four_high_training_score, OAI.SharpRamseyFive.ScoreGeometry.eventually_four_low_training_score, OAI.SharpRamseyFive.ScoreGeometry.eventually_three_training_score and OAI.SharpRamseyFive.ScoreGeometry.eventually_two_score_procedure of this split import it.

Preamble

import Mathlib
import Definitions.Def_OAIQ060R004

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 CellVariance ScoreRegularity
open MeasureTheory ProbabilityTheory PoissonScore
open scoped BigOperators LinearAlgebra.Projectivization Classical NNReal
variable {K V : Type*} [Field K] [AddCommGroup V] [Module K V]
  [Finite K] [FiniteDimensional K V]
  [Fintype (ℙ K V)] [Fintype (ℙ K (Dual K V))]

theorem cell_score_from_integrals {d : ℕ} (hdim : finrank K V=d+1) (hd : 1≤d)
    (hq : (10:ℝ)≤Nat.card K) (X U S : Finset (ℙ K V)) (hSX : S⊆X) (hSU : S⊆U)
    (hS : S.Nonempty) (hret : X.card≤4*S.card)
    {J : Type*} [Fintype J] (C : J→Finset (ℙ K V))
    (hC : (∑j,(C j).card)≤3*S.card) (ia ib : ℙ K V→J)
    (hf : ∀x,ownFraction S (C (ia x)) (C (ib x))≤2/25)
    (T : Finset (ℙ K (Dual K V))) (L : ℝ≥0) (R : ℕ) (b τ CP : ℝ)
    (hτ : 0<τ) (hτhi : 8*τ≤4/5)
    (hdensity : (Nat.card K:ℝ)*(incidences X T:ℝ)≤τ*X.card*T.card)
    (hprod : (Nat.card K:ℝ)^(d+1)*Real.exp (-b)≤(X.card:ℝ)*T.card)
    (hP : 104≤(L:ℝ)*R) (hCP : 0≤CP)
    (hnP : (Nat.card K:ℝ)*((L:ℝ)*R)≤S.card)
    (hbad : ((irregular (d:=d) S C ((L:ℝ)/100)).card:ℝ)≤(S.card:ℝ)/100)
    (he : b+16*((L:ℝ)*R)*τ+Real.log 16+Real.log 3200000000≤((L:ℝ)*R)/2000)
    (hs : 8*((L:ℝ)*R)*τ+Real.log 1600≤(L:ℝ)/1000)
    (hu : 8*((L:ℝ)*R)*τ+Real.log 16≤(L:ℝ)*R)
    (hk : b+8*((L:ℝ)*R)*τ+Real.log 16≤(L:ℝ)*R)
    (hκL : b+8*((L:ℝ)*R)*τ+Real.log 16+Real.log 100≤(91/100:ℝ)*(L:ℝ))
    (hR : (R:ℝ)*Real.exp (-(23/25:ℝ)*(L:ℝ))≤1/10)
    (htrain : (∫ω,((typicalFailures S S (fun x => C (ia x)∪C (ib x)) (exceptional S C)
      (fun x => Real.exp (-(L:ℝ)*(1-ownFraction S (C (ia x)) (C (ib x)))))
      (scale (K:=K) d S.card*Real.exp (-(b+8*((L:ℝ)*R)*τ+Real.log 16))/10) ω).card:ℝ)
      ∂scheduleMeasure (fun _ : S => L*pointStrength S) R)≤(S.card:ℝ)*Real.exp (-(L:ℝ)/1000))
    (hamb : (∫ω,((typicalFailures U S (fun x => C (ia x)∪C (ib x)) (exceptional S C)
      (fun x => Real.exp (-(L:ℝ)*(1-ownFraction S (C (ia x)) (C (ib x)))))
      (scale (K:=K) d S.card*Real.exp (-(b+8*((L:ℝ)*R)*τ+Real.log 16))/10) ω).card:ℝ)
      ∂scheduleMeasure (fun _ : S => L*pointStrength S) R)≤(S.card:ℝ)*Real.exp (CP*((L:ℝ)*R))) :
    Real.exp (-8*((L:ℝ)*R)*τ)/4≤
    (scheduleMeasure (fun _ : S => L*pointStrength S) R).real
      {ω | let W := decoded U S (fun x => C (ia x)∪C (ib x)) (fun x => Real.exp (-(L:ℝ)*(1-ownFraction S (C (ia x)) (C (ib x))))) (emptyTests (exceptional S C) (hyperplaneSupport S) ω).card ω
        (sampleCount ω:ℝ)≤2*((Nat.card K:ℝ)*((L:ℝ)*R)) ∧
        (W.card:ℝ)≤(S.card:ℝ)*Real.exp ((CP+2)*((L:ℝ)*R)) ∧
        (9/10:ℝ)*S.card≤((W∩S).card:ℝ)} := 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/Iteration/CellScoreProcedure.lean (declaration OAI.SharpRamseyFive.ScoreGeometry.cell_score_from_integrals)

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