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Source claim

Let $d_B$ denote the least dimension in which Borsuk's conjecture fails. Theorem 1 of the pinned May 2026 note (p. 1) reads: "Borsuk's conjecture fails in dimension 63. In particular, $d_B\le63$." The abstract on p. 1 describes the construction as a 321-point set $X\subset\mathbb R^{63}$ every subset of which of strictly smaller diameter has at most 5 points; that subset bound is Lemma 6 (p. 5), for the set built in Lemma 4 and Section 5 (pp. 4--5).

The proof conclusion on p. 6 uses

$$\left\lceil\frac{321}{5}\right\rceil=65>64=63+1.$$

If the claimed configuration and subset bound hold, this counting step precludes a partition into 64 smaller-diameter parts. Normalizing a nonzero finite diameter to one preserves that obstruction. The counting step does not independently verify the configuration or subset bound.

Proof pointer and unresolved dependencies

The note's Sections 2--6 (pp. 1--6) contain the finite model, Euclidean representation, dimension reduction, added point, and clique obstruction, recorded here as Lemma 1 (p. 2), Lemma 2 (pp. 2--3), Lemma 3 (pp. 3--4), Lemma 4 (p. 4), Lemma 5 (p. 5) and Lemma 6 (pp. 5--6); the proof of Theorem 1 is on p. 6. Section 7 on that page attributes the finite checks in Lemma 1 to the accompanying script and says it rebuilds the graph from the projective plane $\mathrm{PG}(2,16)$, verifies the strongly regular parameters, builds the partition and checks its degree data, and checks the clique obstruction. These finite facts, their geometric use, and the full proof remain independently unreviewed here. The script, exported certificates, and optional Sage checker have not been executed or audited. No formal certificate of this theorem is established by this entry. Read depth: claims checked; Theorem 1 and its proof were read on pp. 1 and 6, and the lemmas on their own pages.

The source discloses GPT-5.5 Pro assistance. It is public unpublished work; the retained theorem is an attributed claim, not a verified bound in this corpus. borsuk_dimension_63_public_claims records the comparison with later claims and the outstanding review task.

Bears on

• E0505: claims a counterexample in dimension 63 without changing the established disproved status of the general question.