Jenrich (2014), A 64-dimensional two-distance counterexample

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section_7: Jenrich's solo manuscript selects 352 of Bondarenko's 416 G_2(4) vectors, orthogonal to one further vector, so that they span at most 64 dimensions while every smaller-diameter part holds at most five of them; the computational graph facts of Section 6 are taken as given.

section_8: Jenrich's 63-dimensional almost-counterexample: the 320 G_2(4) vectors indexed by C span at most 63 dimensions, and C divides into 64 five-cliques, so the five-point counting bound gives no counterexample in dimension 63.

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Thomas Jenrich, A 64-dimensional two-distance counterexample to Borsuk's conjecture, arXiv:1308.0206v6, 20 August 2014, 7 pages; first submitted 1 August 2013. The copy read for this card is the solo v6 manuscript. Its identity and version were checked on 2026-09-06. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1308.0206), every other right reserved.

Digest and source distinction

The first page explicitly distinguishes this manuscript from the shorter joint paper with Brouwer, which was then submitted to EJC. The subsequently published jenrich_brouwer_2014_borsuk_counterexample has its own card and theorem page. It is not an edition of this solo manuscript.

Section 7, pp. 3--4, gives a 352-vector two-distance configuration in dimension at most 64, with at most five vectors in a smaller-diameter part, hence at least 71 parts; it is recorded at section_7. The same section says the proofs that certain dimension inequalities are equalities are not included. Page 1 mentions the combinatorial computations, Section 6 (p. 3) states the graph facts the program G24CHK checks, and Sections 9--10 (pp. 5--6) describe its implementation. The program has not been executed here. The published theorem is recorded once at theorem_1.

Section 8, on p. 4, gives a 320-vector configuration in dimension at most 63, recorded at section_8. It reports that this core can be partitioned into 64 five-cliques, giving 64 smaller-diameter parts, and explicitly omits that partition's proof. This is a near-counterexample, not a dimension-63 disproof. The borsuk_dimension_63_public_claims compare later public attempts to add a projected point to such a core.

Reading and proof scope

Pages 1--4 were read on the page images, and the statements of Sections 7 and 8 with their setting were checked clause by clause (read depth: claims checked). No full proof reconstruction, program replay, or certificate review is supplied. The computational dependencies and omitted proofs remain visible in this source's separate record.

Bears on

• E0505: Section 7 gives a negative answer in dimension 64, resting on the computer-checked graph facts of Section 6 and on Bondarenko's clique bound, which the manuscript takes as given; Section 8 does not decide dimension 63 and is the 320-point core the public dimension-63 claims extend.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.