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Source. M. Grinsztajn, A 63-dimensional counterexample to Borsuk's conjecture, unpublished note, May 2026, as described on the source card. Lemma 2 is on p. 2, with its consequence and proof on p. 3.

Statement

For the graph $\Gamma$ of Lemma 1 there are vectors $x_v\in\mathbb R^{65}$, $v\in V(\Gamma)$, with $x_u\cdot x_v=90$ if $u=v$, $18$ if $u\sim v$, and $-6$ if $u\not\sim v$. Consequently, for $u\ne v$, $\lVert x_u-x_v\rVert^2$ is $144$ if $u\sim v$ and $192$ if $u\not\sim v$.

The note remarks (p. 3) that this input configuration is two-distance but the final set it builds is not.

Proof pointer

p. 3: with $A$ the adjacency matrix and $J$ the all-one matrix, the matrix $96I+24A-6J$ has eigenvalue $0$ on the all-one line, $576$ on the 65-dimensional $20$-eigenspace and $0$ on the $-4$-eigenspace, by the parameters and multiplicities of Lemma 1, item 1. It is therefore positive semidefinite of rank 65 and is the Gram matrix of the required vectors.

Dependencies and read depth

Depends on Lemma 1, item 1. Read depth: claims checked; the statement and proof were read on pp. 2--3.

Bears on. E0505: the 65-dimensional configuration from which the note's dimension-63 claim (Theorem 1) is cut down.