{"paper":{"title":"Short Second Proof of the Odd-Modulus Directed Torus Hamilton Decomposition Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.CO","authors_text":"Sanghyun Park","submitted_at":"2026-06-19T16:37:39Z","abstract_excerpt":"Let $D_d(m)=\\operatorname{Cay}((\\mathbb Z/m\\mathbb Z)^d,\\{e_1,\\ldots,e_d\\})$, with all generators oriented positively. We give a second proof that $D_d(m)$ decomposes into $d$ directed Hamilton cycles for every $d\\ge 2$ and every odd $m\\ge 3$. The combinatorial core is a fixed-row-sum selection theorem for replicated supports: when each indexed support $A$ is repeated in $m$ identical rows, one can select $\\lfloor |A|/2\\rfloor$ entries from each row so that every column total is a unit modulo $m$. Applied to the Hamilton factors using a chosen coordinate direction, these selections prescribe t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.21583","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.21583/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}