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

Proved

OAI.EuclideanFiveColor.annulus_maps_eventually_forest

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.annulus_maps_eventually_forest of lean/OAI/Geometry/PlaneColoring/PlaneColoring/FullApproximation.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

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

theorem annulus_maps_eventually_forest {c : ℂ → Fin 5} (hc : Measurable c)
    (hw : WeakColoring 5 c) {x : ℂ} {R : ℝ} (hR : 0<R) {r ε : ℕ → ℝ}
    (hr : Tendsto r atTop (𝓝 0)) (hε : Tendsto ε atTop (𝓝 0)) (hpos : ∀n,0<ε n)
    {H E : ℕ → Set ℂ} (hE : Tendsto (fun n => volume (E n)) atTop (𝓝 0))
    (hnear : ∀n z q,q∈H n → dist z q=1 → z∈E n)
    {α : ℝ} (hα : 0<α) (hα' : α<1/5)
    (f : ∀n,C(planeAnnulus x (r n) R,GraphSpace (⊤ : SimpleGraph (Fin 5))))
    (hf : ∀n z,z.val∉H n → (f n z).val=thresholdMap c (ε n) α z.val) :
    ∀ᶠ n in atTop,(crossingGraph (fun z => (f n z).val)
      (annulusInner x (r n) R) (annulusOuter x (r n) R)).IsAcyclic

Definitions. The statement is written with OAI.EuclideanFiveColor.GraphSpace, OAI.EuclideanFiveColor.graphRealization, OAI.EuclideanFiveColor.planeAnnulus and OAI.EuclideanFiveColor.thresholdMap (definitions bundle OAIQ5T4000 of this split); OAI.EuclideanFiveColor.WeakColoring, OAI.EuclideanFiveColor.annulusInner, OAI.EuclideanFiveColor.annulusOuter and OAI.EuclideanFiveColor.crossingGraph (definitions bundle OAIQ5T4002 of this split).

Used by. The proof of OAI.EuclideanFiveColor.fullApprox_exists of this split imports 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 scoped Classical
open Set Filter Topology TopologicalSpace MeasureTheory ContinuousMap
namespace EuclideanFiveColor
theorem annulus_maps_eventually_forest {c : ℂ → Fin 5} (hc : Measurable c)
    (hw : WeakColoring 5 c) {x : ℂ} {R : ℝ} (hR : 0<R) {r ε : ℕ → ℝ}
    (hr : Tendsto r atTop (𝓝 0)) (hε : Tendsto ε atTop (𝓝 0)) (hpos : ∀n,0<ε n)
    {H E : ℕ → Set ℂ} (hE : Tendsto (fun n => volume (E n)) atTop (𝓝 0))
    (hnear : ∀n z q,q∈H n → dist z q=1 → z∈E n)
    {α : ℝ} (hα : 0<α) (hα' : α<1/5)
    (f : ∀n,C(planeAnnulus x (r n) R,GraphSpace (⊤ : SimpleGraph (Fin 5))))
    (hf : ∀n z,z.val∉H n → (f n z).val=thresholdMap c (ε n) α z.val) :
    ∀ᶠ n in atTop,(crossingGraph (fun z => (f n z).val)
      (annulusInner x (r n) R) (annulusOuter x (r n) R)).IsAcyclic := 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/FullApproximation.lean (declaration OAI.EuclideanFiveColor.annulus_maps_eventually_forest)

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