{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:T32QV7VUAKYVWCLLQQWXAHINQW","short_pith_number":"pith:T32QV7VU","schema_version":"1.0","canonical_sha256":"9ef50afeb402b15b096b842d701d0d85bcf724d33bab6ba192cd7fc6a2471288","source":{"kind":"arxiv","id":"2508.04532","version":1},"attestation_state":"computed","paper":{"title":"How are pseudo-$q$-traces related to (co)ends?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","math.RT"],"primary_cat":"math.QA","authors_text":"Bin Gui, Hao Zhang","submitted_at":"2025-08-06T15:13:48Z","abstract_excerpt":"Let $\\mathbb V$ be an $\\mathbb N$-graded $C_2$-cofinite vertex operator algebra (VOA), not necessarily rational or self-dual. Using a special case of the sewing-factorization theorem from [GZ25a], we show that the end $\\mathbb E=\\int_{\\mathbb M\\in\\mathrm{Mod}(\\mathbb V)}\\mathbb M\\otimes_{\\mathbb C}\\mathbb M'$ in $\\mathrm{Mod}(\\mathbb{V}^{\\otimes2})$ (where $\\mathbb{M}'$ is the contragredient module of $\\mathbb{M}$) admits a natural structure of associative $\\mathbb C$-algebra compatible with its $\\mathbb{V}^{\\otimes2}$-module structure. Moreover, we show that a suitable category $\\mathrm{Coh}_"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2508.04532","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2025-08-06T15:13:48Z","cross_cats_sorted":["math-ph","math.MP","math.RT"],"title_canon_sha256":"456697a9ddd9976bea8cfb85344fc3a22053fde2d814d0f31fd1d0a24fc4e413","abstract_canon_sha256":"3622740243d5d973f7616e9bd1d0c2087067a7a4ab054d2ab81e3c8327ec56b4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:49:39.056036Z","signature_b64":"TMuOhaOmxcXFEaSnFVGciSDYYmTjfQZfCeZ05NUNTPCu4nzFn10RZ0i8ZXEGN5FTncd8yTBUSMUTqbZdugr8Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9ef50afeb402b15b096b842d701d0d85bcf724d33bab6ba192cd7fc6a2471288","last_reissued_at":"2026-07-05T11:49:39.055593Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:49:39.055593Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"How are pseudo-$q$-traces related to (co)ends?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","math.RT"],"primary_cat":"math.QA","authors_text":"Bin Gui, Hao Zhang","submitted_at":"2025-08-06T15:13:48Z","abstract_excerpt":"Let $\\mathbb V$ be an $\\mathbb N$-graded $C_2$-cofinite vertex operator algebra (VOA), not necessarily rational or self-dual. Using a special case of the sewing-factorization theorem from [GZ25a], we show that the end $\\mathbb E=\\int_{\\mathbb M\\in\\mathrm{Mod}(\\mathbb V)}\\mathbb M\\otimes_{\\mathbb C}\\mathbb M'$ in $\\mathrm{Mod}(\\mathbb{V}^{\\otimes2})$ (where $\\mathbb{M}'$ is the contragredient module of $\\mathbb{M}$) admits a natural structure of associative $\\mathbb C$-algebra compatible with its $\\mathbb{V}^{\\otimes2}$-module structure. Moreover, we show that a suitable category $\\mathrm{Coh}_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.04532","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/2508.04532/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":"2508.04532","created_at":"2026-07-05T11:49:39.055651+00:00"},{"alias_kind":"arxiv_version","alias_value":"2508.04532v1","created_at":"2026-07-05T11:49:39.055651+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2508.04532","created_at":"2026-07-05T11:49:39.055651+00:00"},{"alias_kind":"pith_short_12","alias_value":"T32QV7VUAKYV","created_at":"2026-07-05T11:49:39.055651+00:00"},{"alias_kind":"pith_short_16","alias_value":"T32QV7VUAKYVWCLL","created_at":"2026-07-05T11:49:39.055651+00:00"},{"alias_kind":"pith_short_8","alias_value":"T32QV7VU","created_at":"2026-07-05T11:49:39.055651+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.19622","citing_title":"One-point functions for $C_2$-cofinite VOAs: pseudo-traces and trace spaces of projective modules","ref_index":13,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/T32QV7VUAKYVWCLLQQWXAHINQW","json":"https://pith.science/pith/T32QV7VUAKYVWCLLQQWXAHINQW.json","graph_json":"https://pith.science/api/pith-number/T32QV7VUAKYVWCLLQQWXAHINQW/graph.json","events_json":"https://pith.science/api/pith-number/T32QV7VUAKYVWCLLQQWXAHINQW/events.json","paper":"https://pith.science/paper/T32QV7VU"},"agent_actions":{"view_html":"https://pith.science/pith/T32QV7VUAKYVWCLLQQWXAHINQW","download_json":"https://pith.science/pith/T32QV7VUAKYVWCLLQQWXAHINQW.json","view_paper":"https://pith.science/paper/T32QV7VU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2508.04532&json=true","fetch_graph":"https://pith.science/api/pith-number/T32QV7VUAKYVWCLLQQWXAHINQW/graph.json","fetch_events":"https://pith.science/api/pith-number/T32QV7VUAKYVWCLLQQWXAHINQW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/T32QV7VUAKYVWCLLQQWXAHINQW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/T32QV7VUAKYVWCLLQQWXAHINQW/action/storage_attestation","attest_author":"https://pith.science/pith/T32QV7VUAKYVWCLLQQWXAHINQW/action/author_attestation","sign_citation":"https://pith.science/pith/T32QV7VUAKYVWCLLQQWXAHINQW/action/citation_signature","submit_replication":"https://pith.science/pith/T32QV7VUAKYVWCLLQQWXAHINQW/action/replication_record"}},"created_at":"2026-07-05T11:49:39.055651+00:00","updated_at":"2026-07-05T11:49:39.055651+00:00"}