Hamilton Decompositions of the Directed 3-Torus

Precise statement

For $D_3(m) = \vec{C}_m \square \vec{C}_m \square \vec{C}_m$, can the full arc set be partitioned into three directed Hamilton cycles for every integer $m \ge 3$?

The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.

What AI did

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

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

Provider: Anthropic / OpenAI · Prompt public: unknown · Independence: unknown

Verification boundary

lean verified statement audited

A Lean 4 formalization accompanies the construction; arXiv preprint.

Correctness: supported · statement fidelity: audited · peer review: none

Timeline

1. 2026-03-25

   arXiv:2603.24708 - Hamilton decompositions of the directed 3-torus

   VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.

Known method families

argument (source-reported)

Source-reported tools: argument.

Independent: unknown · difference confidence: 0

What remains uncertain

VibeMath has not independently audited the mathematical statement, proof, or novelty claim.

• VibeMath has not independently verified the mathematical claim.
• AI-attempt independence and training-data exposure are unknown unless explicitly documented.
• VibeMath has not independently audited the mathematical statement, proof, or novelty claim.