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Statement

There exists a two-distance set $T\subset\mathbb R^{64}$ with $|T|=352$ such that every partition of $T$ into subsets of diameter strictly less than $\operatorname{diam}(T)$ has at least $71$ parts.

Source: Jenrich--Brouwer, published PDF, Theorem 1, p. 3. The preceding paragraph establishes the stronger local property that every smaller-diameter subset of this $T$ has at most five points. Thus the counting bound is $\lceil352/5\rceil=71$.

Since $71>64+1$, rescaling $T$ to diameter one disproves the diameter-one question in dimension 64. This is an upper bound on the first failing dimension, not a proof that dimension 64 is the first one.

Proof pointer and dependencies

The proof is in sections 2--4 and the paragraph preceding Theorem 1, pp. 1--3. Its graph representation uses Bondarenko's $G_2(4)$ construction; the graph structure and equitable partition use the graph-theoretic sources cited there. On p. 3 the authors form a nonzero vector orthogonal to the 352 selected points, placing them in a copy of $\mathbb R^{64}$. The smaller-diameter subset bound then gives the stated partition obstruction.

This is a proof pointer and brief outline. A complete rewritten proof and independent review of the representation, clique bound, partition data, and external graph inputs remain outstanding. The computational qualifications of Jenrich's separate solo manuscript must not be substituted for the published paper's argument or used as its theorem locator.

Bears on

• E0505: a published counterexample in dimension 64, independent of the unreviewed public dimension-63 claims.