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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 M. Grinsztajn, A 63-dimensional counterexample to Borsuk's conjecture, unpublished note, May 2026, as described on the source card.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 6 runs from p. 5 to p. 6; Lemma 5 and its proof are on p. 5.

Statement

Notation as in Lemma 4, with $X=\{x_c:c\in C\}\cup\{p\}$ and $t=(\sqrt{222}-1)/13$. Lemma 5 (p. 5): $X$ has squared diameter $192$. More precisely, for distinct $c,c'\in C$, $\lVert x_c-x_{c'}\rVert^2$ is $144$ if $c\sim c'$ and $192$ if $c\not\sim c'$; and for $c\in C$, $\lVert p-x_c\rVert^2$ is $192-48t$ if $b\sim c$ and $192$ if $b\not\sim c$.

Proof pointer

p. 5: distances among the $x_c$ come from Lemma 2; distances from $p$ come from $\lVert p\rVert^2=78t^2$, Lemma 4 and the equation $78t^2+12t=102$, and $192-48t<192$ since $t>0$. The value 192 is attained because $b$ has 80 neighbors in $C$ and $\lvert C\rvert=320$.

Dependencies and read depth

Depends on Lemma 1, item 4, Lemma 2 and Lemma 4. Read depth: claims checked; the statement and proof were read on p. 5.

Bears on. E0505: identifies which pairs of the note's set are at full diameter, the input to Lemma 6.