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This polynomial records the complete distribution of Morse vectors across all discrete Morse functions on $K$, and is an isomorphism invariant of simplicial complexes.\n  Our main results are the following. \\textbf{(I) The Laplacian Formula}: for any connected graph $G$, $\\ME_G = z_1^{m-n}\\det(z_0z_1\\,I_n + L_G)$,"},"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":"2605.24689","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-23T17:56:04Z","cross_cats_sorted":["math.AT","math.SP"],"title_canon_sha256":"4f360fe1113174bb75cf45f29541b5ef06d3b35c71028358e6ff7b9f97e50b4d","abstract_canon_sha256":"f3571d115410fd27eae90570796a1789d14483be78d6e128a62dd5a9c14f3d7b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-26T01:03:53.432129Z","signature_b64":"LGXApk2FSyfahZTnoyigOldS+zeaEnTwkJo34C4rNqFTlD9sqgeQ5VGKgafz7AxmWndyAoupksQZbcMVivAqAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"11af01d467074323f1e3b02e5cffe53bc51a683f76090d51b8a1c3eefd0a98e9","last_reissued_at":"2026-05-26T01:03:53.431362Z","signature_status":"signed_v1","first_computed_at":"2026-05-26T01:03:53.431362Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On The Morse Ensemble Polynomial Of Simplicial Complexes","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AT","math.SP"],"primary_cat":"math.CO","authors_text":"Chong Zheng","submitted_at":"2026-05-23T17:56:04Z","abstract_excerpt":"We introduce the \\emph{Morse ensemble polynomial} $\\ME_K(z_0,\\ldots,z_d)$ of a finite simplicial complex $K$, defined as the generating function $\\ME_K = \\sum_M \\prod_i z_i^{c_i(M)}$ over all acyclic matchings $M$ on the face poset of $K$, where $c_i(M)$ counts critical $i$-simplices. This polynomial records the complete distribution of Morse vectors across all discrete Morse functions on $K$, and is an isomorphism invariant of simplicial complexes.\n  Our main results are the following. \\textbf{(I) The Laplacian Formula}: for any connected graph $G$, $\\ME_G = z_1^{m-n}\\det(z_0z_1\\,I_n + L_G)$,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.24689","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/2605.24689/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2605.24689","created_at":"2026-05-26T01:03:53.431488+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.24689v1","created_at":"2026-05-26T01:03:53.431488+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.24689","created_at":"2026-05-26T01:03:53.431488+00:00"},{"alias_kind":"pith_short_12","alias_value":"CGXQDVDHA5BS","created_at":"2026-05-26T01:03:53.431488+00:00"},{"alias_kind":"pith_short_16","alias_value":"CGXQDVDHA5BSH4PD","created_at":"2026-05-26T01:03:53.431488+00:00"},{"alias_kind":"pith_short_8","alias_value":"CGXQDVDH","created_at":"2026-05-26T01:03:53.431488+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/CGXQDVDHA5BSH4PDWAXFZ77FHP","json":"https://pith.science/pith/CGXQDVDHA5BSH4PDWAXFZ77FHP.json","graph_json":"https://pith.science/api/pith-number/CGXQDVDHA5BSH4PDWAXFZ77FHP/graph.json","events_json":"https://pith.science/api/pith-number/CGXQDVDHA5BSH4PDWAXFZ77FHP/events.json","paper":"https://pith.science/paper/CGXQDVDH"},"agent_actions":{"view_html":"https://pith.science/pith/CGXQDVDHA5BSH4PDWAXFZ77FHP","download_json":"https://pith.science/pith/CGXQDVDHA5BSH4PDWAXFZ77FHP.json","view_paper":"https://pith.science/paper/CGXQDVDH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.24689&json=true","fetch_graph":"https://pith.science/api/pith-number/CGXQDVDHA5BSH4PDWAXFZ77FHP/graph.json","fetch_events":"https://pith.science/api/pith-number/CGXQDVDHA5BSH4PDWAXFZ77FHP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CGXQDVDHA5BSH4PDWAXFZ77FHP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CGXQDVDHA5BSH4PDWAXFZ77FHP/action/storage_attestation","attest_author":"https://pith.science/pith/CGXQDVDHA5BSH4PDWAXFZ77FHP/action/author_attestation","sign_citation":"https://pith.science/pith/CGXQDVDHA5BSH4PDWAXFZ77FHP/action/citation_signature","submit_replication":"https://pith.science/pith/CGXQDVDHA5BSH4PDWAXFZ77FHP/action/replication_record"}},"created_at":"2026-05-26T01:03:53.431488+00:00","updated_at":"2026-05-26T01:03:53.431488+00:00"}