{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:I7EB5O2NVFGM5NRVQRZRNJMOC5","short_pith_number":"pith:I7EB5O2N","schema_version":"1.0","canonical_sha256":"47c81ebb4da94cceb635847316a58e177c6dc5bf7bc795f6bb3ac0307e04af46","source":{"kind":"arxiv","id":"2608.07599","version":1},"attestation_state":"computed","paper":{"title":"Cycle-Decorated Ribbon Bar Complexes: Cut Factorization and Equivariant Homology","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.CO","authors_text":"Pyuyi Chufeng Huang","submitted_at":"2026-08-06T16:04:04Z","abstract_excerpt":"We introduce bigraded $\\mathfrak S_n$-complexes whose ordered-set-partition bars are decorated by an ordinary and a rooted permutation. Their Hilbert--Euler characteristic is $n!Z_\\alpha(t,q)$, obtained from a two-parameter character of noncommutative symmetric functions. When the composition $\\alpha$ has at most one odd part, simultaneous unique factorization of total decorations gives a canonical splitting \\[\n  C_\\bullet^{t,q}(\\alpha)\\cong\\bigoplus_{\\theta\\in\\mathcal D_n}C_\\bullet(\\gamma_\\alpha(\\theta)) \\] into classical ribbon complexes. We thereby determine every bigraded homology represen"},"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":"2608.07599","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-08-06T16:04:04Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"5261e8bd2307749de540d360132e09dcbe6db5e88e8db2b3559282d5f435df77","abstract_canon_sha256":"616d520ec806b485604d8db5e99449e92a1be660c52254be8761ec36012fa306"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-11T00:15:27.686037Z","signature_b64":"ifkANXI1tORCJq7ZBjLdgynX7pU2udDDXFkwx3MbPP3XyRtUMHxhStkR++bbi1oId8FU6aJQmUHH+DL7OGxhBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"47c81ebb4da94cceb635847316a58e177c6dc5bf7bc795f6bb3ac0307e04af46","last_reissued_at":"2026-08-11T00:15:27.683783Z","signature_status":"signed_v1","first_computed_at":"2026-08-11T00:15:27.683783Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cycle-Decorated Ribbon Bar Complexes: Cut Factorization and Equivariant Homology","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.CO","authors_text":"Pyuyi Chufeng Huang","submitted_at":"2026-08-06T16:04:04Z","abstract_excerpt":"We introduce bigraded $\\mathfrak S_n$-complexes whose ordered-set-partition bars are decorated by an ordinary and a rooted permutation. Their Hilbert--Euler characteristic is $n!Z_\\alpha(t,q)$, obtained from a two-parameter character of noncommutative symmetric functions. When the composition $\\alpha$ has at most one odd part, simultaneous unique factorization of total decorations gives a canonical splitting \\[\n  C_\\bullet^{t,q}(\\alpha)\\cong\\bigoplus_{\\theta\\in\\mathcal D_n}C_\\bullet(\\gamma_\\alpha(\\theta)) \\] into classical ribbon complexes. We thereby determine every bigraded homology represen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.07599","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/2608.07599/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":"2608.07599","created_at":"2026-08-11T00:15:27.684164+00:00"},{"alias_kind":"arxiv_version","alias_value":"2608.07599v1","created_at":"2026-08-11T00:15:27.684164+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.07599","created_at":"2026-08-11T00:15:27.684164+00:00"},{"alias_kind":"pith_short_12","alias_value":"I7EB5O2NVFGM","created_at":"2026-08-11T00:15:27.684164+00:00"},{"alias_kind":"pith_short_16","alias_value":"I7EB5O2NVFGM5NRV","created_at":"2026-08-11T00:15:27.684164+00:00"},{"alias_kind":"pith_short_8","alias_value":"I7EB5O2N","created_at":"2026-08-11T00:15:27.684164+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/I7EB5O2NVFGM5NRVQRZRNJMOC5","json":"https://pith.science/pith/I7EB5O2NVFGM5NRVQRZRNJMOC5.json","graph_json":"https://pith.science/api/pith-number/I7EB5O2NVFGM5NRVQRZRNJMOC5/graph.json","events_json":"https://pith.science/api/pith-number/I7EB5O2NVFGM5NRVQRZRNJMOC5/events.json","paper":"https://pith.science/paper/I7EB5O2N"},"agent_actions":{"view_html":"https://pith.science/pith/I7EB5O2NVFGM5NRVQRZRNJMOC5","download_json":"https://pith.science/pith/I7EB5O2NVFGM5NRVQRZRNJMOC5.json","view_paper":"https://pith.science/paper/I7EB5O2N","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2608.07599&json=true","fetch_graph":"https://pith.science/api/pith-number/I7EB5O2NVFGM5NRVQRZRNJMOC5/graph.json","fetch_events":"https://pith.science/api/pith-number/I7EB5O2NVFGM5NRVQRZRNJMOC5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/I7EB5O2NVFGM5NRVQRZRNJMOC5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/I7EB5O2NVFGM5NRVQRZRNJMOC5/action/storage_attestation","attest_author":"https://pith.science/pith/I7EB5O2NVFGM5NRVQRZRNJMOC5/action/author_attestation","sign_citation":"https://pith.science/pith/I7EB5O2NVFGM5NRVQRZRNJMOC5/action/citation_signature","submit_replication":"https://pith.science/pith/I7EB5O2NVFGM5NRVQRZRNJMOC5/action/replication_record"}},"created_at":"2026-08-11T00:15:27.684164+00:00","updated_at":"2026-08-11T00:15:27.684164+00:00"}