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

Proved

OAI.SharpRamseyFive.ScoreGeometry.low_original_tail

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.low_original_tail of lean/OAI/Combinatorics/RamseyFive/Probability/UniformTails.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 scoped BigOperators LinearAlgebra.Projectivization Classical NNReal):

theorem low_original_tail {d : ℕ} (hdim : finrank K V=d+1) (hd : 2≤d)
    {J : Type} [Fintype J] (S : Finset (ℙ K V)) (C : J→Finset (ℙ K V))
    (hS : S.Nonempty) (a b c : J) (ha : C a⊆S) (hb : C b⊆S) (hc : C c=C a∩C b)
    (F : Finset (ℙ K (Dual K V)))
    (hF : ∀H∈F,Incident x H ∧ H∉exceptional S C)
    (hf : ownFraction S (C a) (C b)≤2/25)
    (hn : (Nat.card K:ℝ)/S.card≤1/100)
    (L : ℝ≥0) (hL : 10000≤(L:ℝ)) (R p h : ℕ) (hp : 0<p) (heven : Even p) (hKR : 200≤R/2)
    (hh : 0<h) (hsize : h≤(R/2)/(2*200))
    (herr : (p:ℝ)*h*(19/20:ℝ)^(h-1)<1/2)
    (hx : x∉irregular (d:=d) S C ((L:ℝ)/100)) (cut : ℝ) (hcut : 1 ≤ cut)
    (hpairs : ∀u:ℝ,0<u → u≤2 →
      ((Finset.univ.filter fun z:WeightedPrograms.DistinctPairs F =>
        u ≤ mass (radialWeight x (outsideAt x S (C a∪C b)) (pointStrength S))
          (pencilLines x F z.1∩pencilLines x F z.2.val)).card:ℝ)*u^200≤
        (scale (K:=K) d S.card)^2*Real.exp (((L:ℝ)*R)/50))
    (hrows : ∀H:F,∀u:ℝ,0<u → u≤2 →
      ((Finset.univ.filter fun H':F => H≠H' ∧ u ≤ mass
        (radialWeight x (outsideAt x S (C a∪C b)) (pointStrength S))
        (pencilLines x F H∩pencilLines x F H')).card:ℝ)*u^200≤
        scale (K:=K) d S.card*Real.exp (((L:ℝ)*R)/50))
    (hdegree : ∀H:F,
      ((Finset.univ.filter fun H':F => H≠H' ∧ 1/(100*(p:ℝ)) ≤ mass
        (radialWeight x (outsideAt x S (C a∪C b)) (pointStrength S))
        (pencilLines x F H∩pencilLines x F H')).card:ℝ)≤
        scale (K:=K) d S.card*Real.exp (-3*((L:ℝ)*R)))
    (hdegreeScalar : 2+scale (K:=K) d S.card*Real.exp (-3*((L:ℝ)*R))≤
        scale (K:=K) d S.card*Real.exp (-((L:ℝ)*R)))
    (hp2 : (p:ℝ)^2≤Real.exp (((L:ℝ)*R)/10))
    (hpairScalar : (pencilAlphabet (Nat.card K) d:ℝ)+
      (dyadFactor (outsideAt x S (C a∪C b)).card (((L:ℝ)*R)/100)*
        (scale (K:=K) d S.card)^2)/(1/(100*(p:ℝ)))^200≤
      (scale (K:=K) d S.card)^2*Real.exp (((L:ℝ)*R)/10))
    (hcost : 40*scale (K:=K) d S.card*Real.exp ((L:ℝ)/100)+
      2*((p+1:ℝ)*lowMomentBudget (pencilAlphabet (Nat.card K) d)
        (dyadFactor (outsideAt x S (C a∪C b)).card (((L:ℝ)*R)/100)*(scale (K:=K) d S.card)^2)
        (dyadFactor (outsideAt x S (C a∪C b)).card (((L:ℝ)*R)/100)*scale (K:=K) d S.card)
        (scale (K:=K) d S.card) cut L p R
        (pencilAlphabet (Nat.card K) (d-1)) (pencilAlphabet (Nat.card K) (d-2)))≤
      scale (K:=K) d S.card*Real.exp (((L:ℝ)*R)/5))
    {κ t : ℝ} (hκ : κ+Real.log 10≤((L:ℝ)*R)/20)
    (hpP : 1000*Real.log (Nat.card K)≤(p:ℝ)*((L:ℝ)*R))
    (ht : scale (K:=K) d S.card*Real.exp (-κ)/10≤t) :
    (scheduleMeasure (fun _ : S => L*pointStrength S) R).real
      {ω | Unsampled x S ω ∧ t < |pointScore S (C a∪C b) F
        (Real.exp (-(L:ℝ)*(1-ownFraction S (C a) (C b)))) ω|} ≤
      (Nat.card K:ℝ)^(-(50:ℝ))

Definitions. The statement is written with OAI.SharpRamseyFive.PoissonScore.mass, OAI.SharpRamseyFive.ProjectiveIncidence.Incident and OAI.SharpRamseyFive.ProjectiveIncidence.RadialLine (definitions bundle OAIQ060R000 of this split); OAI.SharpRamseyFive.ScoreGeometry.Unsampled, OAI.SharpRamseyFive.ScoreGeometry.dyadFactor, OAI.SharpRamseyFive.ScoreGeometry.lowMomentBudget, OAI.SharpRamseyFive.ScoreGeometry.outsideAt, OAI.SharpRamseyFive.ScoreGeometry.pencilAlphabet, OAI.SharpRamseyFive.ScoreGeometry.pointStrength, OAI.SharpRamseyFive.ScoreRegularity.ownFraction, OAI.SharpRamseyFive.ScoreRegularity.scale, OAI.SharpRamseyFive.WeightedPrograms.DistinctPairs and OAI.SharpRamseyFive.WeightedPrograms.flat_HighMomentRecovered_7 (definitions bundle OAIQ060R001 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_low_ambient_tail, OAI_SharpRamseyFive_ScoreGeometry_eventually_four_low_score_integrals, OAI.SharpRamseyFive.ScoreGeometry.eventually_three_ambient_tail_cases and OAI.SharpRamseyFive.ScoreGeometry.eventually_two_score_integrals 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

section
open Module ProjectiveIncidence CellVariance ScoreRegularity PoissonScore MeasureTheory
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))]
  (x : ℙ K V) [Fintype (RadialLine x)]

theorem low_original_tail {d : ℕ} (hdim : finrank K V=d+1) (hd : 2≤d)
    {J : Type} [Fintype J] (S : Finset (ℙ K V)) (C : J→Finset (ℙ K V))
    (hS : S.Nonempty) (a b c : J) (ha : C a⊆S) (hb : C b⊆S) (hc : C c=C a∩C b)
    (F : Finset (ℙ K (Dual K V)))
    (hF : ∀H∈F,Incident x H ∧ H∉exceptional S C)
    (hf : ownFraction S (C a) (C b)≤2/25)
    (hn : (Nat.card K:ℝ)/S.card≤1/100)
    (L : ℝ≥0) (hL : 10000≤(L:ℝ)) (R p h : ℕ) (hp : 0<p) (heven : Even p) (hKR : 200≤R/2)
    (hh : 0<h) (hsize : h≤(R/2)/(2*200))
    (herr : (p:ℝ)*h*(19/20:ℝ)^(h-1)<1/2)
    (hx : x∉irregular (d:=d) S C ((L:ℝ)/100)) (cut : ℝ) (hcut : 1 ≤ cut)
    (hpairs : ∀u:ℝ,0<u → u≤2 →
      ((Finset.univ.filter fun z:WeightedPrograms.DistinctPairs F =>
        u ≤ mass (radialWeight x (outsideAt x S (C a∪C b)) (pointStrength S))
          (pencilLines x F z.1∩pencilLines x F z.2.val)).card:ℝ)*u^200≤
        (scale (K:=K) d S.card)^2*Real.exp (((L:ℝ)*R)/50))
    (hrows : ∀H:F,∀u:ℝ,0<u → u≤2 →
      ((Finset.univ.filter fun H':F => H≠H' ∧ u ≤ mass
        (radialWeight x (outsideAt x S (C a∪C b)) (pointStrength S))
        (pencilLines x F H∩pencilLines x F H')).card:ℝ)*u^200≤
        scale (K:=K) d S.card*Real.exp (((L:ℝ)*R)/50))
    (hdegree : ∀H:F,
      ((Finset.univ.filter fun H':F => H≠H' ∧ 1/(100*(p:ℝ)) ≤ mass
        (radialWeight x (outsideAt x S (C a∪C b)) (pointStrength S))
        (pencilLines x F H∩pencilLines x F H')).card:ℝ)≤
        scale (K:=K) d S.card*Real.exp (-3*((L:ℝ)*R)))
    (hdegreeScalar : 2+scale (K:=K) d S.card*Real.exp (-3*((L:ℝ)*R))≤
        scale (K:=K) d S.card*Real.exp (-((L:ℝ)*R)))
    (hp2 : (p:ℝ)^2≤Real.exp (((L:ℝ)*R)/10))
    (hpairScalar : (pencilAlphabet (Nat.card K) d:ℝ)+
      (dyadFactor (outsideAt x S (C a∪C b)).card (((L:ℝ)*R)/100)*
        (scale (K:=K) d S.card)^2)/(1/(100*(p:ℝ)))^200≤
      (scale (K:=K) d S.card)^2*Real.exp (((L:ℝ)*R)/10))
    (hcost : 40*scale (K:=K) d S.card*Real.exp ((L:ℝ)/100)+
      2*((p+1:ℝ)*lowMomentBudget (pencilAlphabet (Nat.card K) d)
        (dyadFactor (outsideAt x S (C a∪C b)).card (((L:ℝ)*R)/100)*(scale (K:=K) d S.card)^2)
        (dyadFactor (outsideAt x S (C a∪C b)).card (((L:ℝ)*R)/100)*scale (K:=K) d S.card)
        (scale (K:=K) d S.card) cut L p R
        (pencilAlphabet (Nat.card K) (d-1)) (pencilAlphabet (Nat.card K) (d-2)))≤
      scale (K:=K) d S.card*Real.exp (((L:ℝ)*R)/5))
    {κ t : ℝ} (hκ : κ+Real.log 10≤((L:ℝ)*R)/20)
    (hpP : 1000*Real.log (Nat.card K)≤(p:ℝ)*((L:ℝ)*R))
    (ht : scale (K:=K) d S.card*Real.exp (-κ)/10≤t) :
    (scheduleMeasure (fun _ : S => L*pointStrength S) R).real
      {ω | Unsampled x S ω ∧ t < |pointScore S (C a∪C b) F
        (Real.exp (-(L:ℝ)*(1-ownFraction S (C a) (C b)))) ω|} ≤
      (Nat.card K:ℝ)^(-(50:ℝ)) := by
  sorry

end

open Module ProjectiveIncidence CellVariance ScoreRegularity PoissonScore MeasureTheory
open Filter ParameterHierarchy
open scoped BigOperators LinearAlgebra.Projectivization Classical NNReal Topology

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/Probability/UniformTails.lean (declaration OAI.SharpRamseyFive.ScoreGeometry.low_original_tail)

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