Borsuk Conjecture lowest-ever counterexample (N=63)
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Statement
Borsuk's conjecture asked whether every bounded set in can be partitioned into subsets of smaller diameter. It is false in dimension 63: there is a set of 321 points in whose smaller-diameter subsets have at most 5 points, so at least parts are required. The previous record dimension was 64 (Jenrich-Brouwer, 2014), and the first failing dimension remains open for . The construction modifies Bondarenko's two-distance set: a 320-point rank-63 subconfiguration plus one added scaled projected point, which makes the set three-distance - precisely why it was not reachable inside the two-distance framework in which all previous work took place.
Context
Priority: the result was first obtained by Max Grinsztajn with GPT-5.5 Pro assistance, published 26 May 2026 and recorded as the current best bound on Tao's optimization-problems ledger. The same construction was found again independently in August 2026 by Nicholas Konz working with Claude, with a different derivation and a fuller AI disclosure; the two efforts were evidently unaware of each other, and the submitter of this entry surfaced the earlier work themselves after publication. Dimension 63 is the current record; whether Borsuk's conjecture fails for any dimension in 4..62 remains open.
Borsuk's conjecture is a famous named problem with a Wikipedia article and a 90-year history; the conjecture itself was already refuted by Kahn-Kalai in 1993, so what this entry records is the current record for the smallest failing dimension, 64 to 63, a serious but incremental step on a well-known question (the record has moved six times since 1993). Scored level with the Hadamard order-668 construction: both are the current record instance of a famous conjecture rather than the conjecture itself.
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For the first solve, Grinsztajn's README states: "The construction and proof were obtained with assistance from GPT-5.5 Pro", with a dedicated "Disclose GPT assistance" commit; no finer division of labour is given, so the tier is the floor for an unspecific disclosure. The independent August 2026 rediscovery by Nicholas Konz with Claude (Fable 5 and Opus 5) carries a much fuller disclosure - Claude produced the counterexample and an exact certificate over - and would rate ai-discovered on its own, but the entry's tier follows the solve it records, which is the first one.
Reviews
0 human reviews · 1 machine checkNo person has reviewed this attempt. 1 machine check below — a machine check is not human verification.
Machine check · not human verification
machine: correctscope Reproduction by the VibeMathed site
Both derivations reproduced by this site on 12 August 2026, independently of each other. For the first solve (Grinsztajn, May 2026): the repository's exact verifier - pure Python integer arithmetic over F16, read before running - was executed locally and passes all checks: it rebuilds the G2(4) strongly regular graph with parameters (416,100,36,20), the B1/B2/B3/C partition and degree data behind the dimension drop, and the clique obstructions forcing every smaller-diameter subset to size at most 5. The repo's GitHub creation date of 2026-05-26 is not forgeable after the fact, and Terence Tao's optimization-problems ledger (constant 28a) independently credits the 63 bound to Grinsztajn, citing this repository. For the August rediscovery (Konz + Claude): we ran the author's stand-alone verifier against the published 321x63 coordinate file and confirmed affine dimension exactly 63, the squared-distance spectrum (53-sqrt(222))/156, 1/4 and 1/3, and independence number 5 for the diameter graph by Bron-Kerbosch, forcing ceil(321/5) = 65 parts where Borsuk allows 64; the distance-class gap is far wider than any float tolerance. Neither write-up is peer-reviewed; neither is on arXiv.
No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.
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