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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 M. Grinsztajn, A 63-dimensional counterexample to Borsuk's conjecture, unpublished note, May 2026, as described on the source card. Section 4 starts on p. 3; Lemma 3 is on p. 3 and its proof ends on p. 4.

Statement

With $B_1,B_2,B_3,C$ as in Lemma 1 and $x_v$ as in Lemma 2, put $S_i=\sum_{y\in B_i}x_y$ for $i=1,2,3$. Lemma 3 (p. 3) states that the points $x_c$, $c\in C$, lie in a common 63-dimensional subspace of the standard $\mathbb R^{65}$ representation.

The proof identifies the subspace as $W=\operatorname{span}(S_1,S_2,S_3)^{\perp}$ and records along the way that $S_i\cdot S_i=12288$, $S_i\cdot S_j=-6144$ for $i\ne j$, and $\dim\operatorname{span}(S_1,S_2,S_3)=2$. The note identifies $W$ with $\mathbb R^{63}$ from then on (p. 4).

Proof pointer

pp. 3--4: each $c\in C$ has 8 neighbors and 24 non-neighbors in each $B_i$, so $x_c\cdot S_i=0$. The degree data inside and between the blocks give the Gram matrix of $S_1,S_2,S_3$, whose eigenvalues $0$, $18432$, $18432$ show that the span is two-dimensional.

Dependencies and read depth

Depends on Lemma 1, items 3 and 4, and Lemma 2. Read depth: claims checked; the statement and proof were read on pp. 3--4.

Bears on. E0505: the reduction from dimension 65 to 63 in the note's claim (Theorem 1).