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Statement

For a finite set of diameter $D$, its compatibility graph joins two points when their distance is strictly smaller than $D$ (p. 7).

Proposition 6.1 (p. 7). The compatibility graph of the set $X=\{x_c:c\in C\}\cup\{z\}$ of display (26) (p. 6) is isomorphic to the induced subgraph $\Gamma[C\cup\{v\}]$ under the map $x_c\mapsto c$, $z\mapsto v$.

Here $\Gamma$ is the $G_2(4)$ graph, strongly regular with parameters $(416,100,36,20)$ (p. 3); $C$ is the 320-vertex part of the equitable partition $B_1\sqcup B_2\sqcup B_3\sqcup C$ (p. 4); $v\in B_1$ is the fixed vertex of Section 5 (p. 5); and $X$ has diameter $\sqrt8$ (p. 6), so compatibility means squared distance below 8.

Source. Yibo Ji, An AI Generated Counterexample to Borsuk Problem in Dimension 63, arXiv:2608.12561v1 (12 August 2026), withdrawn by version 2 of 14 August 2026; Proposition 6.1 on p. 7. The source card records the withdrawal and provenance.

Read depth. Claims checked: the statement and the definitions it uses were read clause by clause on the PDF. The distance identities it rests on were not independently checked.

Proof pointer

Between two core points the squared distance is 6 for adjacent labels and 8 otherwise (display (8), p. 3); between $z$ and $x_c$ it is $8-2t$, below 8, for $c$ adjacent to $v$ and 8 otherwise (display (25), p. 6). The map is a bijection because $v\notin C$ (p. 7).

Dependencies

The Gram-matrix realization of $\Gamma$ (Section 2.2, p. 3), the equitable partition attributed to Jenrich and Brouwer (Section 2.3, p. 4), and the added point given by Lemma 4.1 (Section 5, pp. 5--6).

Bears on

• E0505: only through Theorem 6.2, whose proof turns a subset of smaller diameter into a clique of $\Gamma$ by this identification.