An independent exact recheck of the n = 6 non-transitive dice theorem (a recheck of a result by others)

T. Easlick (arXiv:2610.40321, 29 Oct 2026, unrefereed) studies n-sided dice with faces in {1,...,n} and the directed graph Gn of pairs with the minimal winning advantage; Theorem 1 states that for n = 6 there are exactly two uniform minimal non-transitive cycles of maximum length 6. This is an independent recheck of a result by others. We claim nothing new. What was rechecked (Tier C: exact integer and Fraction arithmetic, two independent implementations). Graph statistics for n = 3..10 (n = 10: 92378 dice, 157,244,770 edges, 10588 SCCs, one nontrivial SCC of 81791 vertices and 147,620,052 edges); the edge-reversing involution fn; no directed cycle for n = 3, 4, 5; for n = 6 exactly two 6-cycles with every edge exactly 19/36, six 3-cycles, no cycle of length 4, 5 or 7, and SCC labels equal to the paper's Appendix E; the paper's printed cycles for n = 7, 8, 9, 10 (lengths 14, 29, 10, 110) verify exactly. A cycle longer than n exists for each n = 7..10, so Conjecture 1 is open only for n ≥ 11 under the reading of the proof of Theorem 1 (and false at n = 7 if 'maximum length n' only means that an n-cycle exists). A randomized search (Tier X) found exact-checked cycles of every length 3..28 in G7. Audit. A separate AI agent instance (not a human) reproduced these results with its own C code (ACCEPT_WITH_CORRECTIONS; six corrections applied). Limitations. Nothing is claimed for n ≥ 11; the vertex lists of the n ≥ 7 SCCs in the paper's appendices were compared by size and count only; the maximin context controls were not recomputed by the auditor; the preregistration was committed in the same commit as the results (self-attested); the prior-art search was limited; no Lean; no human audit. The paper's full text is third-party and is not redistributed.

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Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-11-01
DOI
https://doi.org/10.5281/zenodo.23863825
Primary Topic
Probability and Statistical Research
Type
preprint
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preprint

An independent exact recheck of the n = 6 non-transitive dice theorem (a recheck of a result by others)

Xavier Callens
1 links
Zenodo (CERN European Organization for Nuclear Research)
Probability and Statistical Research
preprint

An independent exact recheck of the n = 6 non-transitive dice theorem (a recheck of a result by others)

Xavier Callens
preprint en

Abstract

T. Easlick (arXiv:2610.40321, 29 Oct 2026, unrefereed) studies n-sided dice with faces in {1,...,n} and the directed graph Gn of pairs with the minimal winning advantage; Theorem 1 states that for n = 6 there are exactly two uniform minimal non-transitive cycles of maximum length 6. This is an independent recheck of a result by others. We claim nothing new. What was rechecked (Tier C: exact integer and Fraction arithmetic, two independent implementations). Graph statistics for n = 3..10 (n = 10: 92378 dice, 157,244,770 edges, 10588 SCCs, one nontrivial SCC of 81791 vertices and 147,620,052 edges); the edge-reversing involution fn; no directed cycle for n = 3, 4, 5; for n = 6 exactly two 6-cycles with every edge exactly 19/36, six 3-cycles, no cycle of length 4, 5 or 7, and SCC labels equal to the paper's Appendix E; the paper's printed cycles for n = 7, 8, 9, 10 (lengths 14, 29, 10, 110) verify exactly. A cycle longer than n exists for each n = 7..10, so Conjecture 1 is open only for n ≥ 11 under the reading of the proof of Theorem 1 (and false at n = 7 if 'maximum length n' only means that an n-cycle exists). A randomized search (Tier X) found exact-checked cycles of every length 3..28 in G7. Audit. A separate AI agent instance (not a human) reproduced these results with its own C code (ACCEPT_WITH_CORRECTIONS; six corrections applied). Limitations. Nothing is claimed for n ≥ 11; the vertex lists of the n ≥ 7 SCCs in the paper's appendices were compared by size and count only; the maximin context controls were not recomputed by the auditor; the preregistration was committed in the same commit as the results (self-attested); the prior-art search was limited; no Lean; no human audit. The paper's full text is third-party and is not redistributed.

Zenodo (CERN European Organization for Nuclear Research)
Probability and Statistical Research
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