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

Proved

OAI.SharpRamseyFive.ProjectiveRestriction.eventually_three_flat_finite_predictor

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.ProjectiveRestriction.eventually_three_flat_finite_predictor of lean/OAI/Combinatorics/RamseyFive/Geometry/ThreeFiniteOriented.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.SharpRamseyFive.ProjectiveRestriction, with open Module ProjectiveIncidence ProjectiveTraining ScoreGeometry FiniteEntropy, open DyadicLifts ReverseCap Filter ParameterHierarchy, open scoped Classical LinearAlgebra.Projectivization NNReal Topology):

theorem eventually_three_flat_finite_predictor {η : ℝ} (hη : 0<η) (hη' : η<1/10)
    (Cb : ℝ) (hCb : 0≤Cb) :
    ∀ᶠ σ : ℝ in atTop,∀ (D b τ : ℝ) (R : ℕ) (L₀ : ℝ≥0),
    ∀ (q : ℕ) (K V : Type) [Field K] [AddCommGroup V] [Module K V]
      [Finite K] [CharP K q] [FiniteDimensional K V] [Fintype (ℙ K V)] [Fintype (ℙ K (Dual K V))],
    ∀ (A : Submodule K V) (hA : finrank K A=4)
      (X UX : Finset (ℙ K V)) (T UT : Finset (ℙ K (Dual K V))),
      finrank K V≤5 → Nat.card K=q → Real.exp σ=q →
      Range η σ D R → (L₀:ℝ)=L η σ D → 0≤b → b≤Cb*D*σ^(6*beta η) →
      0<τ → τ≤σ^(-800*beta η) → X.Nonempty → T.Nonempty → X⊆UX → T⊆UT →
      (Nat.card K:ℝ)*(incidences X T:ℝ)≤τ*X.card*T.card →
      (Nat.card K:ℝ)^(finrank K V)*Real.exp (-b)≤(X.card:ℝ)*T.card →
      (X.card:ℝ)≤100*(X∩flatPoints A).card →
      ∃k≤Nat.log 2 ((Nat.card K)^(finrank K V-finrank K A)),
        let pred := threeFlatFinitePredictor A hA σ UX UT τ (P η σ D R) R L₀ k
        let p := pred.output X T
        p none≤3*Real.exp (-(Nat.card K:ℝ)) ∧
        (∀W,0<p (some W)→W⊆UX ∧ (W.card:ℝ)≤(X.card:ℝ)*Real.exp (10*P η σ D R) ∧
          (9/100000:ℝ)*X.card≤(W∩X).card) ∧
        (∀t m,pred.encoded X T t=some m→pred.cost t m≤
          30000*(Nat.card K:ℝ)*(P η σ D R)*
            (Real.log ((UX.card:ℝ)/X.card)+Real.log ((UT.card:ℝ)/T.card)+(P η σ D R)))

Definitions. The statement is written with OAI.SharpRamseyFive.FiniteEntropy.Law and OAI.SharpRamseyFive.ScoreGeometry.FinitePredictor (definitions bundle OAIQ060R000 of this split); OAI.SharpRamseyFive.ParameterHierarchy.L, OAI.SharpRamseyFive.ParameterHierarchy.P, OAI.SharpRamseyFive.ParameterHierarchy.Range, OAI.SharpRamseyFive.ParameterHierarchy.beta and OAI.SharpRamseyFive.ProjectiveTraining.flatPoints (definitions bundle OAIQ060R001 of this split); OAI.SharpRamseyFive.ProjectiveIncidence.incidences (definitions bundle OAIQ060R002 of this split); OAI.SharpRamseyFive.ProjectiveRestriction.threeFlatFinitePredictor and OAI.SharpRamseyFive.ScoreGeometry.FinitePredictor.output (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

attribute [-instance] _root_.OAI.SharpRamseyFive.ProjectiveIncidence.orientedNodeTapeFintype _root_.OAI.SharpRamseyFive.ProjectiveIncidence.orientedPivotMessageFintype _root_.OAI.SharpRamseyFive.ProjectiveIncidence.orientedPivotTreeMessageFintype _root_.OAI.SharpRamseyFive.ProjectiveIncidence.orientedPivotTreeTapeFintype
set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

namespace SharpRamseyFive.ProjectiveRestriction
open Module ProjectiveIncidence ProjectiveTraining ScoreGeometry FiniteEntropy
open DyadicLifts ReverseCap Filter ParameterHierarchy
open scoped Classical LinearAlgebra.Projectivization NNReal Topology
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 eventually_three_flat_finite_predictor {η : ℝ} (hη : 0<η) (hη' : η<1/10)
    (Cb : ℝ) (hCb : 0≤Cb) :
    ∀ᶠ σ : ℝ in atTop,∀ (D b τ : ℝ) (R : ℕ) (L₀ : ℝ≥0),
    ∀ (q : ℕ) (K V : Type) [Field K] [AddCommGroup V] [Module K V]
      [Finite K] [CharP K q] [FiniteDimensional K V] [Fintype (ℙ K V)] [Fintype (ℙ K (Dual K V))],
    ∀ (A : Submodule K V) (hA : finrank K A=4)
      (X UX : Finset (ℙ K V)) (T UT : Finset (ℙ K (Dual K V))),
      finrank K V≤5 → Nat.card K=q → Real.exp σ=q →
      Range η σ D R → (L₀:ℝ)=L η σ D → 0≤b → b≤Cb*D*σ^(6*beta η) →
      0<τ → τ≤σ^(-800*beta η) → X.Nonempty → T.Nonempty → X⊆UX → T⊆UT →
      (Nat.card K:ℝ)*(incidences X T:ℝ)≤τ*X.card*T.card →
      (Nat.card K:ℝ)^(finrank K V)*Real.exp (-b)≤(X.card:ℝ)*T.card →
      (X.card:ℝ)≤100*(X∩flatPoints A).card →
      ∃k≤Nat.log 2 ((Nat.card K)^(finrank K V-finrank K A)),
        let pred := threeFlatFinitePredictor A hA σ UX UT τ (P η σ D R) R L₀ k
        let p := pred.output X T
        p none≤3*Real.exp (-(Nat.card K:ℝ)) ∧
        (∀W,0<p (some W)→W⊆UX ∧ (W.card:ℝ)≤(X.card:ℝ)*Real.exp (10*P η σ D R) ∧
          (9/100000:ℝ)*X.card≤(W∩X).card) ∧
        (∀t m,pred.encoded X T t=some m→pred.cost t m≤
          30000*(Nat.card K:ℝ)*(P η σ D R)*
            (Real.log ((UX.card:ℝ)/X.card)+Real.log ((UT.card:ℝ)/T.card)+(P η σ D R))) := by
  sorry
end SharpRamseyFive.ProjectiveRestriction

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/ThreeFiniteOriented.lean (declaration OAI.SharpRamseyFive.ProjectiveRestriction.eventually_three_flat_finite_predictor)

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