Chromatic number of the plane is 6 or 7, OpenAI claims (Lean)

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Key facts

• Manuscript (abstract, verbatim): ‘We prove that every coloring of the Euclidean plane with five colors has a monochromatic unit-distance pair, with no regularity assumption on the color classes. Consequently, the chromatic number of the plane is either six or seven.’ 62 pages, dated 23 Sept 2026. Theorem 1.1 is proved in ZFC; ‘The remaining alternatives six and seven are unresolved’
• Structure (paper): ‘a transfer from unrestricted proper colorings to a measurable condition, and a geometric obstruction to five labels under that condition. The transfer holds for every finite number of colors.’ Because of the de Bruijn–Erdős compactness theorem, ‘an unrestricted lower bound can also be established without first displaying a finite graph witnessing it’; the final step reuses the Moser spindle
• Before: Nelson (1950) and Isbell gave the bounds 4 and 7; Aubrey de Grey’s 2018 finite unit-distance graph (1,581 vertices after correction) raised the lower bound to 5, later shrunk by Heule (553 vertices) and Parts (509); for measurable colour classes Falconer had proved at least 5 colours (as recounted in the paper)
• Lean (OpenAI scope note lean/docs/158.md): ‘The formalized results prove that five colors do not suffice and that seven colors do suffice. The lower bound applies to arbitrary colorings, with no measurability or continuity assumption.’ Both challenges ‘accepted’ in the oai-math-revised comparator run (25–26 Oct). Caveat recorded there: the EuclideanFiveColor configuration let the comparator check the definition ProperColoring by type only; OpenAI emptied that list on 26 Oct. Carlos Toledo’s audit had flagged that this one definition ‘carries the whole conclusion’, ran a strict body-by-body comparison, and later lists EuclideanFiveColor as certified after the change
• No explicit witness: Toledo’s audit notes that ‘the finite non-5-colorable unit-distance graph is obtained by compactness and never displayed’, while certifying the paper’s finite Moser-spindle certificate in exact arithmetic
• Reactions: Joshua Zelinsky (Bluesky, 6 Oct): ‘it doesn’t look like the method is a direction that the prior lit used to my knowledge. This is far beyond merely building on existing methods’. Dmitry Rybin (X, 7 Oct, about 261k views): ‘I was trying to read the OpenAI proof that chromatic number of a plane is >= 6. But it is totally unbelievable alien math?’, adding that the model found that any K-colouring is equivalent to a ‘weakly measurable’ one. Gabriel Gaster (X, 9 Oct) pointed to its mix of ‘Pontryagin duality, ergodic theory, and amenable groups’. Gil Kalai (7 Oct): ‘The new manuscript raises the lower bound to six’. Thomas Bloom lists it under Erdős problem #508, noting ‘This proof has been formalised’
• Follow-ups: Koshelev, Kupavskii and Voronov (arXiv 2610.18907, 11 Oct): ‘Using the planar character-law rigidity theorem of OpenAI, we prove that every finite proper coloring of R³ can be replaced by a proper Lebesgue-measurable coloring with the same number of colors’, so the ordinary and measurable chromatic numbers of R³ are equal; they used ChatGPT 6 Astra to extract the main ideas of OpenAI’s preprint. Henry Bass’s GitHub repository extends the transfer theorem to R^n with AI-assisted preprints and Lean