Grinsztajn (2026), public dimension-63 Borsuk claim

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lemma_1: The note's computer-checked facts: the 416-vertex G_2(4) graph is strongly regular with parameters (416,100,36,20) and clique number 5, and a fixed isotropic point splits it into three 32-vertex blocks and a 320-vertex rest with stated degree data.

lemma_2: The 416 vertices of the G_2(4) graph have vectors in R^65 with inner products 90, 18 or -6, so squared distances are 144 between adjacent and 192 between non-adjacent vertices.

lemma_3: The vectors x_c for the 320 vertices c in C are orthogonal to the three block sums S_1, S_2, S_3, which span a plane, so they lie in a common 63-dimensional subspace W of R^65.

lemma_4: For b in B_1 the projection z_b = x_b - S_1/32 lies in W, has squared norm 78, and has the same inner products 18 or -6 with each x_c as x_b; the note then adds p = t z_b with t = (sqrt(222)-1)/13 to get a 321-point set.

lemma_5: In the note's set X, squared distances are 144 or 192 between points x_c and 192 - 48t or 192 from the added point p, according to adjacency in the G_2(4) graph, so X has squared diameter 192.

lemma_6: Every subset of the note's 321-point set X whose diameter is strictly smaller than that of X has at most 5 points, by reduction to the clique number 5 of the G_2(4) graph.

theorem_1: Grinsztajn's May 2026 note claims a 321-point set in R^63 with at most five points in any subset of strictly smaller diameter.

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Max Grinsztajn, A 63-dimensional counterexample to Borsuk's conjecture, May 2026, 6-page public unpublished note. The copy read for this card is the PDF of the pinned repository, commit cdcdbeac2e692b8641218c70ce9f414522e125e5. The GitHub commit record dates that commit to 2026-05-27T09:49:17Z. The PDF's identity at that commit was checked. No notice is printed in the note, and the hosting repository (https://github.com/maaxgrin/borsuk-63-counterexample, read 2026-10-02) carries no LICENSE file and shows no license in its sidebar; the term is unstated.

Claim and provenance

The abstract and theorem_1 claim a 321-point set in $\mathbb R^{63}$ whose smaller-diameter subsets have at most five points. The note begins with a 320-point core of the $G_2(4)$ Euclidean configuration and proposes adjoining one projected and rescaled point. Its p. 6 disclosure attributes the construction and proof to work assisted by GPT-5.5 Pro.

This May source predates Ji's August submission. It is not merely corroboration of Ji. The public project and later priority acknowledgments are compared at borsuk_dimension_63_public_claims. No peer-reviewed publication or independent mathematical acceptance is established by this entry.

Reading and proof scope

All six pages were read: the theorem and its counting conclusion (pp. 1 and 6), Lemmas 1--6 and the proofs of Lemmas 2--6 (pp. 2--6; Lemma 1 has no hand proof), the finite-verification description and the AI disclosure (p. 6). The pinned README lists a deterministic Python verifier, exported DIMACS certificates, a Sage checker, and a reported GitHub Actions workflow. Those are the project's verification claims. No code was run, certificates audited, or build reproduced here. The full mathematical proof has not been independently reviewed. The result pages therefore record source claims and proof pointers at read depth claims checked, with zero verified-result credit.

Results

• Theorem 1 (p. 1; proof p. 6): Borsuk's conjecture fails in dimension 63.
• Lemma 1 (p. 2): the finite facts about the $G_2(4)$ graph that the note attributes to its script, among them the parameters $(416,100,36,20)$, clique number 5, and the blocks $B_1,B_2,B_3$ and the 320-vertex set $C$ with their degree data.
• Lemma 2 (pp. 2--3): the standard representation in $\mathbb R^{65}$ with squared distances 144 and 192.
• Lemma 3 (pp. 3--4): the 320 points of $C$ lie in a 63-dimensional subspace $W$.
• Lemma 4 (p. 4), with the definitions on p. 5: the projected vertex $z_b\in W$ and the added point $p=\frac{\sqrt{222}-1}{13}z_b$, giving 321 points.
• Lemma 5 (p. 5): the set has squared diameter 192, with the full-diameter pairs identified by non-adjacency.
• Lemma 6 (pp. 5--6): every subset of strictly smaller diameter has at most 5 points.

Bears on

• E0505: Theorem 1, through Lemmas 1--6, claims a counterexample in dimension 63, one below the Jenrich--Brouwer dimension 64 the note cites; the claim is unpublished and unreviewed, and the general question was already disproved. The claim is recorded at its claim page.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.