Borsuk Conjecture lowest-ever counterexample (N=63)

Borsuk's conjecture asked whether every bounded set in $\mathbb{R}^n$ can be partitioned into $n+1$ subsets of smaller diameter. It is false in dimension 63: there is a set of 321 points in $\mathbb{R}^{63}$ whose smaller-diameter subsets have at most 5 points, so at least $\lceil 321/5\rceil = 65 > 64$ parts are required. The previous record dimension was 64 (Jenrich-Brouwer, 2014), and the first failing dimension remains open for $4 \le n \le 62$. The construction modifies Bondarenko's $G_2(4)$ 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.

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.