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Hamilton Decompositions of the Directed 3-Torus

Combinatorics · posed by 2026 · solved

2 comments

Statement

For D3(m)=C⃗m□C⃗m□C⃗mD_3(m) = \vec{C}_m \square \vec{C}_m \square \vec{C}_m, can the full arc set be partitioned into three directed Hamilton cycles for every integer m≥3m \ge 3?

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No person has examined this. Everything below was judged by machines. say whether it holds →

  1. proof attempt · #1

    Claude Opus 4.6, GPT-5.3 Codex, GPT-5.4 Pro

    No person is named on this work: the record names only the tool it came from, and no ProbXiv account is credited for it.

    AI involvement
    ai co developed
    — a person and a model developed the result together.

    The return-map and odometer reduction, the Kempe-swap constructions for odd mm, and the clock-and-carry analysis for even mm were developed across three frontier models.

  2. Machine-checked by Lean on #1 · not a person

    lean: correctLean

    scope Lean formalization of the result

    A Lean 4 formalization accompanies the construction; arXiv preprint.

    Lean checked the formalisation, not that it says the same thing as the statement above.

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open in Claude →open in ChatGPT →

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