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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 M. Grinsztajn, A 63-dimensional counterexample to Borsuk's conjecture, unpublished note, May 2026, as described on the source card.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 5 runs from p. 4 to p. 5: Lemma 4 and its proof are on p. 4, and the definitions of $p$ and $X$ are on p. 5.

Statement

Notation as in Lemma 3. Lemma 4 (p. 4): for a choice of $b\in B_1$, the vector $z_b=x_b-\frac1{32}S_1$ lies in $W$, satisfies $\lVert z_b\rVert^2=78$, and for every $c\in C$ has $z_b\cdot x_c=18$ if $b\sim c$ and $-6$ if $b\not\sim c$.

The added point (p. 5). The note sets

$$t=\frac{\sqrt{222}-1}{13},\qquad p=tz_b,$$

so that $t$ is the positive root of $78t^2+12t=102$, and defines $X=\{x_c:c\in C\}\cup\{p\}$. It shows that $p$ is none of the $x_c$ (those have squared norm 90, while $\lVert p\rVert^2=78t^2\ne90$), so $X\subset\mathbb R^{63}$ and $\lvert X\rvert=321$.

Proof pointer

p. 4: $x_b\cdot S_1=384=\frac1{32}S_1\cdot S_1$ and, for $i=2,3$, $x_b\cdot S_i=-192=\frac1{32}S_1\cdot S_i$, so $z_b$ is orthogonal to $S_1,S_2,S_3$; the norm follows from the same values, and $S_1\cdot x_c=0$ (Lemma 3) gives $z_b\cdot x_c=x_b\cdot x_c$, which Lemma 2 evaluates.

Dependencies and read depth

Depends on Lemma 1, item 4, Lemma 2 and Lemma 3. Read depth: claims checked; Lemma 4, its proof and the definitions of $t$, $p$ and $X$ were read on pp. 4--5.

Bears on. E0505: the 321st point of the set that the note's claim (Theorem 1) concerns.