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

Proved

OAI.SharpRamseyFive.ScoreGeometry.eventually_high_original_tail_retained

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.eventually_high_original_tail_retained of lean/OAI/Combinatorics/RamseyFive/Geometry/UniformHighTail.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 PoissonScore MeasureTheory, open Filter ParameterHierarchy, open scoped BigOperators LinearAlgebra.Projectivization Classical NNReal Topology):

theorem eventually_high_original_tail_retained {η : ℝ} (hη : 0<η) (hη' : η<1/10)
    (Cb : ℝ) (hCb : 0≤Cb) :
    ∀ᶠ σ : ℝ in atTop,∀ (D b₀ τ : ℝ) (R : ℕ) (L₀ : ℝ≥0),
    ∀ (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))],
    ∀ (x : ℙ K V) [Fintype (RadialLine x)],
    ∀ {J : Type} [Fintype J] (S : Finset (ℙ K V)) (C : J→Finset (ℙ K V)) (a b c : J)
      (F : Finset (ℙ K (Dual K V))) (t : ℝ),
      finrank K V=5 → (Nat.card K:ℝ)=Real.exp σ →
      Range η σ D R → (L₀:ℝ)=L η σ D → 0≤b₀ → b₀≤Cb*D*σ^(6*beta η) →
      τ≤σ^(-200*beta η) → S.Nonempty → C a⊆S → C b⊆S → C c=C a∩C b →
      (S.card:ℝ)≤10*Real.exp (5*σ/2) →
      (Nat.card K:ℝ)^2*Real.exp (P η σ D R/20000)≤(S.card:ℝ) →
      (Nat.card K:ℝ)/S.card≤1/100 → ownFraction S (C a) (C b)≤2/25 →
      (∀H∈F,Incident x H ∧ H∉exceptional S C) →
      x∉irregular (d:=4) S C ((L₀:ℝ)/100) →
      ∀ (Lines : Finset (Submodule K V)) (Q : Finset (ℙ K V)),
      (∀l : RadialLine x,l.val∈Lines) → x∈Q →
      (∀i∈boundedOverlapDyads x (outsideAt x S (C a∪C b)) (pointStrength S) 2,
        x∉GlobalRadial.badCenters S (fun _ => C a∪C b) (pointStrength S)
          (((pointStrength S:ℝ)*2^i)/2) Lines Q
          ((Nat.card K:ℝ)^4/S.card^2*Real.exp (P η σ D R/100)/
            (((pointStrength S:ℝ)*2^i)/2)^100)) →
      ((∑z : WeightedPrograms.DistinctPairs F,WeightedPrograms.strength (pencilLines x F)
        (radialWeight x (outsideAt x S (C a∪C b)) (pointStrength S)) z^200)≤
        dyadFactor (outsideAt x S (C a∪C b)).card (P η σ D R/100)*
          (geometryScale (Nat.card K) S.card)^2) →
      x∉GlobalRadial.badCenters S (fun _ => C a∪C b) (pointStrength S)
        ((1/(100*(momentOrder σ (P η σ D R):ℝ)))/2) Lines Q 1 →
      geometryScale (Nat.card K) S.card*Real.exp (-(b₀+8*P η σ D R*τ+Real.log 16))/10≤t →
      (scheduleMeasure (fun _ : S => L₀*pointStrength S) R).real
        {ω | Unsampled x S ω ∧ t < |pointScore S (C a∪C b) F
          (Real.exp (-(L₀:ℝ)*(1-ownFraction S (C a) (C b)))) ω|} ≤
        (Nat.card K:ℝ)^(-(50:ℝ))+
      ∫ω,1-HighMoment.allTrunc R 5000 ω
        ∂batchMeasure (fun i : Fin R×RadialLine x => L₀*
          radialWeight x (outsideAt x S (C a∪C b)) (pointStrength S) i.2)

Definitions. The statement is written with OAI.SharpRamseyFive.PoissonScore.batchMeasure, OAI.SharpRamseyFive.ProjectiveIncidence.Incident and OAI.SharpRamseyFive.ProjectiveIncidence.RadialLine (definitions bundle OAIQ060R000 of this split); OAI.SharpRamseyFive.GlobalRadial.badCenters, OAI.SharpRamseyFive.ParameterHierarchy.L, OAI.SharpRamseyFive.ParameterHierarchy.P, OAI.SharpRamseyFive.ParameterHierarchy.Range, OAI.SharpRamseyFive.ParameterHierarchy.beta, OAI.SharpRamseyFive.ScoreGeometry.Unsampled, OAI.SharpRamseyFive.ScoreGeometry.boundedOverlapDyads, OAI.SharpRamseyFive.ScoreGeometry.dyadFactor, OAI.SharpRamseyFive.ScoreGeometry.geometryScale, OAI.SharpRamseyFive.ScoreGeometry.momentOrder, OAI.SharpRamseyFive.ScoreGeometry.outsideAt, OAI.SharpRamseyFive.ScoreGeometry.pointStrength, OAI.SharpRamseyFive.ScoreRegularity.ownFraction, OAI.SharpRamseyFive.WeightedPrograms.DistinctPairs, OAI.SharpRamseyFive.WeightedPrograms.flat_HighMomentRecovered_7 and OAI.SharpRamseyFive.WeightedPrograms.strength (definitions bundle OAIQ060R001 of this split); OAI.SharpRamseyFive.HighMoment.allTrunc (definitions bundle OAIQ060R002 of this split); OAI.SharpRamseyFive.PoissonScore.scheduleMeasure, OAI.SharpRamseyFive.ScoreGeometry.pencilLines, OAI.SharpRamseyFive.ScoreGeometry.pointScore, OAI.SharpRamseyFive.ScoreGeometry.radialWeight 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_score_integrals and OAI.SharpRamseyFive.ScoreGeometry.eventually_four_high_training_score 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 PoissonScore MeasureTheory
open Filter ParameterHierarchy
open scoped BigOperators LinearAlgebra.Projectivization Classical NNReal Topology

theorem eventually_high_original_tail_retained {η : ℝ} (hη : 0<η) (hη' : η<1/10)
    (Cb : ℝ) (hCb : 0≤Cb) :
    ∀ᶠ σ : ℝ in atTop,∀ (D b₀ τ : ℝ) (R : ℕ) (L₀ : ℝ≥0),
    ∀ (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))],
    ∀ (x : ℙ K V) [Fintype (RadialLine x)],
    ∀ {J : Type} [Fintype J] (S : Finset (ℙ K V)) (C : J→Finset (ℙ K V)) (a b c : J)
      (F : Finset (ℙ K (Dual K V))) (t : ℝ),
      finrank K V=5 → (Nat.card K:ℝ)=Real.exp σ →
      Range η σ D R → (L₀:ℝ)=L η σ D → 0≤b₀ → b₀≤Cb*D*σ^(6*beta η) →
      τ≤σ^(-200*beta η) → S.Nonempty → C a⊆S → C b⊆S → C c=C a∩C b →
      (S.card:ℝ)≤10*Real.exp (5*σ/2) →
      (Nat.card K:ℝ)^2*Real.exp (P η σ D R/20000)≤(S.card:ℝ) →
      (Nat.card K:ℝ)/S.card≤1/100 → ownFraction S (C a) (C b)≤2/25 →
      (∀H∈F,Incident x H ∧ H∉exceptional S C) →
      x∉irregular (d:=4) S C ((L₀:ℝ)/100) →
      ∀ (Lines : Finset (Submodule K V)) (Q : Finset (ℙ K V)),
      (∀l : RadialLine x,l.val∈Lines) → x∈Q →
      (∀i∈boundedOverlapDyads x (outsideAt x S (C a∪C b)) (pointStrength S) 2,
        x∉GlobalRadial.badCenters S (fun _ => C a∪C b) (pointStrength S)
          (((pointStrength S:ℝ)*2^i)/2) Lines Q
          ((Nat.card K:ℝ)^4/S.card^2*Real.exp (P η σ D R/100)/
            (((pointStrength S:ℝ)*2^i)/2)^100)) →
      ((∑z : WeightedPrograms.DistinctPairs F,WeightedPrograms.strength (pencilLines x F)
        (radialWeight x (outsideAt x S (C a∪C b)) (pointStrength S)) z^200)≤
        dyadFactor (outsideAt x S (C a∪C b)).card (P η σ D R/100)*
          (geometryScale (Nat.card K) S.card)^2) →
      x∉GlobalRadial.badCenters S (fun _ => C a∪C b) (pointStrength S)
        ((1/(100*(momentOrder σ (P η σ D R):ℝ)))/2) Lines Q 1 →
      geometryScale (Nat.card K) S.card*Real.exp (-(b₀+8*P η σ D R*τ+Real.log 16))/10≤t →
      (scheduleMeasure (fun _ : S => L₀*pointStrength S) R).real
        {ω | Unsampled x S ω ∧ t < |pointScore S (C a∪C b) F
          (Real.exp (-(L₀:ℝ)*(1-ownFraction S (C a) (C b)))) ω|} ≤
        (Nat.card K:ℝ)^(-(50:ℝ))+
      ∫ω,1-HighMoment.allTrunc R 5000 ω
        ∂batchMeasure (fun i : Fin R×RadialLine x => L₀*
          radialWeight x (outsideAt x S (C a∪C b)) (pointStrength S) i.2) := 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/UniformHighTail.lean (declaration OAI.SharpRamseyFive.ScoreGeometry.eventually_high_original_tail_retained)

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