Not evaluated
The Euclidean plane cannot be colored with five colors (family 158)
The gist
On October 6, 2026, OpenAI announced that the Euclidean plane cannot be coloured with five colours, using an unreleased internal model.
DiscoverIA status on October 12, 2026: not evaluated, meaning no third-party evaluation found.
AI draft, not reviewedWhat has not been reviewed?
Text written by an AI (DiscoverIA pipeline), on . The status and its sources come from the registry; only the explanatory text has not been reviewed. A reported error is corrected, then the text is reviewed.
Cite this entry
A single status snapshot since Oct 12, 2026: a citation always points to a dated snapshot.
Cite snapshot 1, Oct 12, 2026
APA citation with the dated link
BibTeX @misc, key discoveria-oai-fam-158-euclidean-plane-cannot-be-colored-five-r1
@misc{discoveria-oai-fam-158-euclidean-plane-cannot-be-colored-five-r1,
author = {{DiscoverIA}},
title = {The Euclidean plane cannot be colored with five colors (family 158)},
howpublished = {DiscoverIA registry, entry oai-fam-158-euclidean-plane-cannot-be-colored-five},
note = {Status “Not evaluated”, snapshot 1, recorded on October 12, 2026. New result},
year = {2026},
url = {https://discoveria.ai/entry/oai-fam-158-euclidean-plane-cannot-be-colored-five#releve-1},
urldate = {2026-10-12}
}Dated link opens this snapshot in the history
To understand ≈ 30 s reading time, estimated
What it changes
If the manuscript were correct, it would bring, according to its title, the impossibility of colouring the Euclidean plane with five colours. A colouring of the plane, in the usual sense of this problem, gives each point a colour so that two points at distance exactly one always have different colours.
What remains unclear
The Lean (a language in which a computer checks every step of a proof) was supplied by OpenAI itself, and its tracking file formalization.yaml marks it “unchecked”. No third-party verification is recorded, and the sources read specify neither the colouring condition used nor the scope of the Lean.
For experts
Sources, method, key facts, status history, report an error
Status
Not evaluated
Recorded on · github.com · 158.md
Step 1 of 4 on the scale: not evaluated. Next step: under review.
- Announced
- Recordednot evaluated
- nextTo follow
What this status means
Source reading : Source read word for word (level 3 of 3)
The result is declared neither right nor wrong.
To move to “Under review”: a public review under way. To “Peer reviewed”: a journal publication or a review by named experts.
Sources
- Main sourcegithub.com · CONTENTS.mdlink only, reading not recorded
- Repositorygithub.com · The-Euclidean-plane-is…link only, reading not recorded
- Lean proofgithub.com · 158.mdpage read word for word
- Lean proofgithub.com · EuclideanFiveColor.leanlink only, reading not recorded
- Lean proofgithub.com · PlaneColoring.leanlink only, reading not recorded
Method
The openai/math repository contains one manuscript, “The Euclidean plane is not five-colorable”, dated September 23, 2026 and filed under the section “Combinatorics” (overview.tex). The file lean/docs/158.md describes the scope of the Lean formalisation, with 2 Comparator statements. The files are lean/ComparatorChallenges/EuclideanFiveColor.lean and lean/ComparatorChallenges/PlaneColoring.lean. The family is listed among the sources of formalization.yaml, and history.md reports no revision of this manuscript. DiscoverIA status: not evaluated, meaning that no third-party evaluation has been found in the sources read. Set on October 12, 2026 from github.com (page read, quotation checked word for word: “describes the scope of the Lean formalization”).
Limits
No named third party: the recorded checks come from the claimant or from a platform tool. The claimant supplies a Lean proof; as of this date, DiscoverIA has found no independent verification of this proof or of the result.
Key facts
- Claimed by
- OpenAI (unreleased internal model)
announced on Oct 6, 2026 - Nature
- new result
- Field
- Mathematics (Combinatorics)
- In openai/math
- family 158, section “Combinatorics”, 1 manuscript dated Sep 23, 2026
- Lean proof
- supplied by the claimant, marked “unchecked” (not reviewed) in its repository
- Revisions
- none reported by the claimant
- Checked by
- claimant (Lean supplied, not checked by a third party)
- Sources
- 5 : 1 read word for word, 4 links only
- This entry
- AI draft, not reviewed
Status history
Only one observation so far.
- 01
First observation:Not evaluatedCurrent status
Family with a Lean proof supplied by the claimant. No third-party check read; the source marks this proof as not reviewed (“unchecked”).
Source: github.comread word for word
An observation is added; none is ever erased.
In the same set
openai/math: 719 manuscripts in 372 families from an unreleased internal model · entry 153 of 372, by family number
- Not evaluated : Pointwise multiple ergodic averages for mixing transformations (family 154)
- Not evaluated : A counterexample to periodic tiling in dimension three (family 155)
- Not evaluated : Borsuk's conjecture fails in dimension nine (family 156)
- Not evaluated : The Euclidean plane cannot be colored with five colors (family 158)
- Not evaluated : Erdős's reciprocal-sum conjecture and quasipolynomial Szemerédi bounds (family 159)
- Not evaluated : Superexponential van der Waerden numbers (family 160)
- Not evaluated : Counterexamples to Sidorenko's conjecture and the forcing conjecture (family 161)
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