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

Proved

OAI.SharpRamseyFive.ScoreGeometry.actual_high_radial_moment

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.actual_high_radial_moment of lean/OAI/Combinatorics/RamseyFive/Entropy/High.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 ProjectiveTraining GlobalRadial WeightedPrograms, open scoped BigOperators LinearAlgebra.Projectivization Classical NNReal):

theorem actual_high_radial_moment {J : Type} [Fintype J]
    (hdim : finrank K V=5) (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)
    (hKR : 200≤R/2) (hh : 0<h) (hsize : h≤(R/2-5000)/(2*200))
    (herr : (p:ℝ)*h*(19/20:ℝ)^(h-1)<1/2)
    (hx : x∉irregular (d:=4) S C ((L:ℝ)/100)) (cut : ℝ) (hcut : 1 ≤ cut)
    (hcap : (S.card:ℝ)≤(Nat.card K:ℝ)^3)
    (hchi : (208*2^198:ℝ)≤Real.exp (2*(((L:ℝ)*R)/100)))
    (hmax : (pointStrength S:ℝ)*((Nat.card K:ℝ)+1)≤2)
    (Lines : Finset (Submodule K V)) (hLines : ∀l : RadialLine x,l.val∈Lines)
    (Q : Finset (ℙ K V)) (hxQ : x∈Q)
    (hgood : ∀i∈boundedOverlapDyads x (outsideAt x S (C a∪C b)) (pointStrength S) 2,
      x∉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 (((L:ℝ)*R)/100)/
          (((pointStrength S:ℝ)*2^i)/2)^100))
    (hPow : (∑z : DistinctPairs F,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 (((L:ℝ)*R)/100)*
        (geometryScale (Nat.card K) S.card)^2)
    (hlocal : (pencilAlphabet (Nat.card K) 3:ℝ)*((R:ℝ)*(L:ℝ))^5000*
        (((pointStrength S:ℝ)*((Nat.card K:ℝ)+1))^4900*
          (2^200*((Nat.clog 2 (outsideAt x S (C a∪C b)).card:ℝ)+1)*
            ((Nat.card K:ℝ)^4/S.card^2*Real.exp (((L:ℝ)*R)/100))))≤
      geometryScale (Nat.card K) S.card)
    (hcost : (40*geometryScale (Nat.card K) S.card*Real.exp ((L:ℝ)/100)+
      geometryScale (Nat.card K) S.card)+2*((p+1:ℝ)*
        pencilHighBudget (Nat.card K) (outsideAt x S (C a∪C b)).card S.card
          (((L:ℝ)*R)/100) cut L p R)≤
      geometryScale (Nat.card K) S.card*Real.exp (((L:ℝ)*R)/5))
    (hp2 : (p:ℝ)^2≤Real.exp (((L:ℝ)*R)/10))
    (hA : 2≤geometryScale (Nat.card K) S.card*Real.exp (-((L:ℝ)*R)))
    (hscale : 16*((Nat.card K:ℝ)+1)≤
      (geometryScale (Nat.card K) S.card*Real.exp (-((L:ℝ)*R)))*(1/(100*(p:ℝ)))^2)
    (hstrong : x∉badCenters S (fun _ => C a∪C b) (pointStrength S)
      ((1/(100*(p:ℝ)))/2) Lines Q 1)
    (hpairScalar : (pencilAlphabet (Nat.card K) 4:ℝ)+
      (dyadFactor (outsideAt x S (C a∪C b)).card (((L:ℝ)*R)/100)*
        (geometryScale (Nat.card K) S.card)^2)/(1/(100*(p:ℝ)))^200≤
      (geometryScale (Nat.card K) S.card)^2*Real.exp (((L:ℝ)*R)/10))
    (own : F→Fin R→Bool) :
    (∫ω,HighMoment.allTrunc R 5000 ω*(∑H:F,scoreTerm (pencilLines x F H)
      (Real.exp (-(L:ℝ)*(1-ownFraction S (C a) (C b)))) (own H)
      (fun r e => ω (r,e)))^p
      ∂batchMeasure (fun i : Fin R×RadialLine x => L*
        radialWeight x (outsideAt x S (C a∪C b)) (pointStrength S) i.2))≤
      (geometryScale (Nat.card K) S.card)^p*Real.exp (-(1/10:ℝ)*(p*((L:ℝ)*R)))

Definitions. The statement is written with OAI.SharpRamseyFive.PoissonScore.batchMeasure, OAI.SharpRamseyFive.PoissonScore.scoreTerm, OAI.SharpRamseyFive.ProjectiveIncidence.Incident and OAI.SharpRamseyFive.ProjectiveIncidence.RadialLine (definitions bundle OAIQ060R000 of this split); OAI.SharpRamseyFive.GlobalRadial.badCenters, OAI.SharpRamseyFive.ScoreGeometry.boundedOverlapDyads, OAI.SharpRamseyFive.ScoreGeometry.dyadFactor, OAI.SharpRamseyFive.ScoreGeometry.geometryScale, OAI.SharpRamseyFive.ScoreGeometry.outsideAt, OAI.SharpRamseyFive.ScoreGeometry.pencilAlphabet, OAI.SharpRamseyFive.ScoreGeometry.pencilHighBudget, 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.ScoreGeometry.pencilLines, 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 proof of OAI.SharpRamseyFive.ScoreGeometry.eventually_high_original_tail_retained of this split imports 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 ProjectiveTraining GlobalRadial WeightedPrograms
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 actual_high_radial_moment {J : Type} [Fintype J]
    (hdim : finrank K V=5) (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)
    (hKR : 200≤R/2) (hh : 0<h) (hsize : h≤(R/2-5000)/(2*200))
    (herr : (p:ℝ)*h*(19/20:ℝ)^(h-1)<1/2)
    (hx : x∉irregular (d:=4) S C ((L:ℝ)/100)) (cut : ℝ) (hcut : 1 ≤ cut)
    (hcap : (S.card:ℝ)≤(Nat.card K:ℝ)^3)
    (hchi : (208*2^198:ℝ)≤Real.exp (2*(((L:ℝ)*R)/100)))
    (hmax : (pointStrength S:ℝ)*((Nat.card K:ℝ)+1)≤2)
    (Lines : Finset (Submodule K V)) (hLines : ∀l : RadialLine x,l.val∈Lines)
    (Q : Finset (ℙ K V)) (hxQ : x∈Q)
    (hgood : ∀i∈boundedOverlapDyads x (outsideAt x S (C a∪C b)) (pointStrength S) 2,
      x∉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 (((L:ℝ)*R)/100)/
          (((pointStrength S:ℝ)*2^i)/2)^100))
    (hPow : (∑z : DistinctPairs F,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 (((L:ℝ)*R)/100)*
        (geometryScale (Nat.card K) S.card)^2)
    (hlocal : (pencilAlphabet (Nat.card K) 3:ℝ)*((R:ℝ)*(L:ℝ))^5000*
        (((pointStrength S:ℝ)*((Nat.card K:ℝ)+1))^4900*
          (2^200*((Nat.clog 2 (outsideAt x S (C a∪C b)).card:ℝ)+1)*
            ((Nat.card K:ℝ)^4/S.card^2*Real.exp (((L:ℝ)*R)/100))))≤
      geometryScale (Nat.card K) S.card)
    (hcost : (40*geometryScale (Nat.card K) S.card*Real.exp ((L:ℝ)/100)+
      geometryScale (Nat.card K) S.card)+2*((p+1:ℝ)*
        pencilHighBudget (Nat.card K) (outsideAt x S (C a∪C b)).card S.card
          (((L:ℝ)*R)/100) cut L p R)≤
      geometryScale (Nat.card K) S.card*Real.exp (((L:ℝ)*R)/5))
    (hp2 : (p:ℝ)^2≤Real.exp (((L:ℝ)*R)/10))
    (hA : 2≤geometryScale (Nat.card K) S.card*Real.exp (-((L:ℝ)*R)))
    (hscale : 16*((Nat.card K:ℝ)+1)≤
      (geometryScale (Nat.card K) S.card*Real.exp (-((L:ℝ)*R)))*(1/(100*(p:ℝ)))^2)
    (hstrong : x∉badCenters S (fun _ => C a∪C b) (pointStrength S)
      ((1/(100*(p:ℝ)))/2) Lines Q 1)
    (hpairScalar : (pencilAlphabet (Nat.card K) 4:ℝ)+
      (dyadFactor (outsideAt x S (C a∪C b)).card (((L:ℝ)*R)/100)*
        (geometryScale (Nat.card K) S.card)^2)/(1/(100*(p:ℝ)))^200≤
      (geometryScale (Nat.card K) S.card)^2*Real.exp (((L:ℝ)*R)/10))
    (own : F→Fin R→Bool) :
    (∫ω,HighMoment.allTrunc R 5000 ω*(∑H:F,scoreTerm (pencilLines x F H)
      (Real.exp (-(L:ℝ)*(1-ownFraction S (C a) (C b)))) (own H)
      (fun r e => ω (r,e)))^p
      ∂batchMeasure (fun i : Fin R×RadialLine x => L*
        radialWeight x (outsideAt x S (C a∪C b)) (pointStrength S) i.2))≤
      (geometryScale (Nat.card K) S.card)^p*Real.exp (-(1/10:ℝ)*(p*((L:ℝ)*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/Entropy/High.lean (declaration OAI.SharpRamseyFive.ScoreGeometry.actual_high_radial_moment)

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