OAI.EuclideanFiveColor.binary_centers_separated: intermediate lemma from OpenAI, The Euclidean plane cannot be colored with five colors

Proved

OAI.EuclideanFiveColor.binary_centers_separated

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.EuclideanFiveColor.no_proper_five_coloring, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, The Euclidean plane cannot be colored with five colors, OpenAI Math Release, September 23, 2026, result family 158, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.EuclideanFiveColor.binary_centers_separated of lean/OAI/Geometry/PlaneColoring/PlaneColoring/BinaryPalettes.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.EuclideanFiveColor, with open Set Filter Topology MeasureTheory):

lemma binary_centers_separated {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c) (hw : WeakColoring k c) :
    ∃ δ > 0, ∀ (x z : ℂ) (i j : Fin k), BinaryCenter c hc x i j → BinaryCenter c hc z i j →
      x ≠ z → δ ≤ dist x z

Definitions. The statement is written with OAI.EuclideanFiveColor.BinaryCenter (definitions bundle OAIQ5T4000 of this split); OAI.EuclideanFiveColor.WeakColoring (definitions bundle OAIQ5T4002 of this split).

Used by. The proofs of OAI.EuclideanFiveColor.ae_not_binary_circle and OAI.EuclideanFiveColor.threeCenter_compact_finite of this split import it.

Preamble

import Mathlib
import Definitions.Def_OAIQ5T4002

Formal statement

section

attribute [-instance] _root_.OAI.EuclideanFiveColor.BoundedMeasurable.product_measurable_vadd
set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

noncomputable section
open Set Filter Topology MeasureTheory
namespace EuclideanFiveColor
lemma binary_centers_separated {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c) (hw : WeakColoring k c) :
    ∃ δ > 0, ∀ (x z : ℂ) (i j : Fin k), BinaryCenter c hc x i j → BinaryCenter c hc z i j →
      x ≠ z → δ ≤ dist x z := by
  sorry
end EuclideanFiveColor
end

end OAI
end

Source

OpenAI, The Euclidean plane cannot be colored with five colors, OpenAI Math Release, September 23, 2026, https://github.com/openai/math (result family 158; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/paper.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Geometry/PlaneColoring/PlaneColoring/BinaryPalettes.lean (declaration OAI.EuclideanFiveColor.binary_centers_separated)

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