OAI.PlanarUnitDistances.IncidenceBook.constants: intermediate lemma from OpenAI, A power saving for planar unit distances

Proved

OAI.PlanarUnitDistances.IncidenceBook.constants

by dbenbenn · Oct 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.PlanarUnitDistances.main, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, A power saving for planar unit distances, OpenAI Math Release, September 23, 2026, result family 167, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.PlanarUnitDistances.IncidenceBook.constants of lean/OAI/Geometry/UnitDistances/UnitDistances/Profiles.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.PlanarUnitDistances.IncidenceBook, with open Filter FiniteLaw FiniteStarCoupling RareLabels RareMoments ScoreBook, open scoped Classical):

theorem constants (s : Sequence) :
    ∃s₀ C L : ℝ,0 < s₀ ∧ 0 < C ∧ 0 < L ∧
      ∀a : Params,2*a.epsilon ≤ s₀ → ∀ᶠi in atTop,∃R : Realization s a i,R.Valid C L

Definitions. The statement is written with OAI.PlanarUnitDistances.BudgetParameters.Params, OAI.PlanarUnitDistances.IncidenceBook.Realization, OAI.PlanarUnitDistances.IncidenceSequence.Sequence and OAI.PlanarUnitDistances.IncidenceBook.Realization.Valid (definitions bundle OAIQ2Un000 of this split).

Used by. The proof of OAI.PlanarUnitDistances.IncidenceBand.choose of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIQ2Un000

Formal statement

section

attribute [-instance] _root_.OAI.PlanarUnitDistances.IncidenceDiagonal.Data.instFieldK _root_.OAI.PlanarUnitDistances.IncidenceDiagonal.Data.instIsGaloisRatK _root_.OAI.PlanarUnitDistances.IncidenceDiagonal.Data.instNumberFieldK
set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

namespace PlanarUnitDistances.IncidenceBook
open Filter FiniteLaw FiniteStarCoupling RareLabels RareMoments ScoreBook
  IncidenceSequence IncidenceScale ActualScoreBook LevelSnapshot BudgetParameters
open scoped Classical
noncomputable section
variable (s : Sequence) (a : Params) (i : ℕ)

variable {s a i}

theorem constants (s : Sequence) :
    ∃s₀ C L : ℝ,0 < s₀ ∧ 0 < C ∧ 0 < L ∧
      ∀a : Params,2*a.epsilon ≤ s₀ → ∀ᶠi in atTop,∃R : Realization s a i,R.Valid C L := by
  sorry
end
end PlanarUnitDistances.IncidenceBook

end OAI
end

Source

OpenAI, A power saving for planar unit distances, OpenAI Math Release, September 23, 2026, https://github.com/openai/math (result family 167; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/paper.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Geometry/UnitDistances/UnitDistances/Profiles.lean (declaration OAI.PlanarUnitDistances.IncidenceBook.constants)

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