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

Proved

OAI.PlanarUnitDistances.IncidenceBand.choose

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.IncidenceBand.choose of lean/OAI/Geometry/UnitDistances/UnitDistances/Tails.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.PlanarUnitDistances.IncidenceBand, with open Filter FiniteLaw FiniteStarCoupling IncidenceSequence IncidenceScale IncidenceBook, open scoped Classical):

theorem choose (s : Sequence) : ∃c : ℝ,0<c ∧ ∀δ : ℝ,0<δ →
    ∃a : Params,∃b : ℝ,0<b ∧ ∀ᶠi in atTop,∃R : Realization s a i,∃J : ℝ,
      c*scale s i/2≤J ∧ b≤event (WedgeLaw.law (graph s i)) (band R J δ)

Definitions. The statement is written with OAI.PlanarUnitDistances.BudgetParameters.Params, OAI.PlanarUnitDistances.FiniteLaw.event, OAI.PlanarUnitDistances.GeometricLaw.Plane, OAI.PlanarUnitDistances.IncidenceBand.band, OAI.PlanarUnitDistances.IncidenceBook.Realization, OAI.PlanarUnitDistances.IncidenceScale.graph, OAI.PlanarUnitDistances.IncidenceScale.scale, OAI.PlanarUnitDistances.IncidenceSequence.Sequence and OAI.PlanarUnitDistances.WedgeLaw.law (definitions bundle OAIQ2Un000 of this split).

Used by. The proof of OAI.PlanarUnitDistances.main 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.IncidenceBand
open Filter FiniteLaw FiniteStarCoupling IncidenceSequence IncidenceScale IncidenceBook
  IncidenceRare IncidenceHeights ActualProfiles ProfileBudget BudgetParameters
open scoped Classical
noncomputable section
variable {s : Sequence} {a : Params} {i : ℕ}

theorem choose (s : Sequence) : ∃c : ℝ,0<c ∧ ∀δ : ℝ,0<δ →
    ∃a : Params,∃b : ℝ,0<b ∧ ∀ᶠi in atTop,∃R : Realization s a i,∃J : ℝ,
      c*scale s i/2≤J ∧ b≤event (WedgeLaw.law (graph s i)) (band R J δ) := by
  sorry
end
end PlanarUnitDistances.IncidenceBand

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/Tails.lean (declaration OAI.PlanarUnitDistances.IncidenceBand.choose)

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