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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2606.21583","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-19T16:37:39Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"6f3698f3ff74f552afda0ba0a7fc9c4a674241b99e2ab524904157e300e5a5f2","abstract_canon_sha256":"4226ad58a0c26276ec8d762623b06d896be913a5f24ccfed10a5a98a051ce851"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-23T01:13:15.353436Z","signature_b64":"u3BzoAXXffw42TQjY/Li0T762x/4919zpMpDNs7oO3IYO9eCgNcdXaJApooSv7U7D3r1D4vAGPnxgCMT2BJ7AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"31232b94ae379191d5a41ffaaace4a8319238748353caf58e632408ac478e497","last_reissued_at":"2026-06-23T01:13:15.352932Z","signature_status":"signed_v1","first_computed_at":"2026-06-23T01:13:15.352932Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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