{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:1995:2QN6RQ6QJOCEBI3NMBYD7XLQ67","short_pith_number":"pith:2QN6RQ6Q","canonical_record":{"source":{"id":"q-alg/9505019","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"q-alg","submitted_at":"1995-05-17T17:23:59Z","cross_cats_sorted":["hep-th","math.QA"],"title_canon_sha256":"c9d3644c2413126203064e490af48bbb7232c41b4cc39681d4fe65dd29f98fb0","abstract_canon_sha256":"298b46b1aa8162e2264823d5457856b2da6c909909b6c55731b7ea54635ac9ab"},"schema_version":"1.0"},"canonical_sha256":"d41be8c3d04b8440a36d60703fdd70f7c5284ed66d3d359c1313f2dc5fefdb73","source":{"kind":"arxiv","id":"q-alg/9505019","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"q-alg/9505019","created_at":"2026-07-04T15:09:05Z"},{"alias_kind":"arxiv_version","alias_value":"q-alg/9505019v1","created_at":"2026-07-04T15:09:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.q-alg/9505019","created_at":"2026-07-04T15:09:05Z"},{"alias_kind":"pith_short_12","alias_value":"2QN6RQ6QJOCE","created_at":"2026-07-04T15:09:05Z"},{"alias_kind":"pith_short_16","alias_value":"2QN6RQ6QJOCEBI3N","created_at":"2026-07-04T15:09:05Z"},{"alias_kind":"pith_short_8","alias_value":"2QN6RQ6Q","created_at":"2026-07-04T15:09:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:1995:2QN6RQ6QJOCEBI3NMBYD7XLQ67","target":"record","payload":{"canonical_record":{"source":{"id":"q-alg/9505019","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"q-alg","submitted_at":"1995-05-17T17:23:59Z","cross_cats_sorted":["hep-th","math.QA"],"title_canon_sha256":"c9d3644c2413126203064e490af48bbb7232c41b4cc39681d4fe65dd29f98fb0","abstract_canon_sha256":"298b46b1aa8162e2264823d5457856b2da6c909909b6c55731b7ea54635ac9ab"},"schema_version":"1.0"},"canonical_sha256":"d41be8c3d04b8440a36d60703fdd70f7c5284ed66d3d359c1313f2dc5fefdb73","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:09:05.752803Z","signature_b64":"2sfaIe0VZb53uHSAJJnk0mr0cHtquYJKFj2zGVWi5e8MkqbOYa5Sz+damOkzq/HlBvd/JHnS8PNUuqb1cp4RAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d41be8c3d04b8440a36d60703fdd70f7c5284ed66d3d359c1313f2dc5fefdb73","last_reissued_at":"2026-07-04T15:09:05.752426Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:09:05.752426Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"q-alg/9505019","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T15:09:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FU4h5dcrgqFsaNcC4Juaa6mC50mCofxHzfIQ+B4QyJCGBt3YV0e0x2nVZtLCV5x11oXpV+WGzA5VeBBLU2GEBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T23:32:08.142105Z"},"content_sha256":"d88397824a7a9690ffff1139f9f5caaa738781c31e9b4b7b9e668838989683c4","schema_version":"1.0","event_id":"sha256:d88397824a7a9690ffff1139f9f5caaa738781c31e9b4b7b9e668838989683c4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:1995:2QN6RQ6QJOCEBI3NMBYD7XLQ67","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A theory of tensor products for module categories for a vertex operator algebra, IV","license":"","headline":"","cross_cats":["hep-th","math.QA"],"primary_cat":"q-alg","authors_text":"Yi-Zhi Huang","submitted_at":"1995-05-17T17:23:59Z","abstract_excerpt":"This is the fourth part of a series of papers developing a tensor product theory of modules for a vertex operator algebra. In this paper, We establish the associativity of $P(z)$-tensor products for nonzero complex numbers $z$ constructed in Part III of the present series under suitable conditions. The associativity isomorphisms constructed in this paper are analogous to associativity isomorphisms for vector space tensor products in the sense that it relates the tensor products of three elements in three modules taken in different ways. The main new feature is that they are controlled by the d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"q-alg/9505019","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/q-alg/9505019/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T15:09:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jxWBszs03PWuH4IAaRM7dYG9VcMR5S/BTq1wPW9v18rxXLgogIGGPAv/2YQfdsXDOUs2c4RQtqYQAhBEeqJmCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T23:32:08.142915Z"},"content_sha256":"f60e0e00dd81f94d806d8889ff1bcb4aed03ce739b8eca1e5e6dc58a4e55c312","schema_version":"1.0","event_id":"sha256:f60e0e00dd81f94d806d8889ff1bcb4aed03ce739b8eca1e5e6dc58a4e55c312"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2QN6RQ6QJOCEBI3NMBYD7XLQ67/bundle.json","state_url":"https://pith.science/pith/2QN6RQ6QJOCEBI3NMBYD7XLQ67/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2QN6RQ6QJOCEBI3NMBYD7XLQ67/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-10T23:32:08Z","links":{"resolver":"https://pith.science/pith/2QN6RQ6QJOCEBI3NMBYD7XLQ67","bundle":"https://pith.science/pith/2QN6RQ6QJOCEBI3NMBYD7XLQ67/bundle.json","state":"https://pith.science/pith/2QN6RQ6QJOCEBI3NMBYD7XLQ67/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2QN6RQ6QJOCEBI3NMBYD7XLQ67/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1995:2QN6RQ6QJOCEBI3NMBYD7XLQ67","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"298b46b1aa8162e2264823d5457856b2da6c909909b6c55731b7ea54635ac9ab","cross_cats_sorted":["hep-th","math.QA"],"license":"","primary_cat":"q-alg","submitted_at":"1995-05-17T17:23:59Z","title_canon_sha256":"c9d3644c2413126203064e490af48bbb7232c41b4cc39681d4fe65dd29f98fb0"},"schema_version":"1.0","source":{"id":"q-alg/9505019","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"q-alg/9505019","created_at":"2026-07-04T15:09:05Z"},{"alias_kind":"arxiv_version","alias_value":"q-alg/9505019v1","created_at":"2026-07-04T15:09:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.q-alg/9505019","created_at":"2026-07-04T15:09:05Z"},{"alias_kind":"pith_short_12","alias_value":"2QN6RQ6QJOCE","created_at":"2026-07-04T15:09:05Z"},{"alias_kind":"pith_short_16","alias_value":"2QN6RQ6QJOCEBI3N","created_at":"2026-07-04T15:09:05Z"},{"alias_kind":"pith_short_8","alias_value":"2QN6RQ6Q","created_at":"2026-07-04T15:09:05Z"}],"graph_snapshots":[{"event_id":"sha256:f60e0e00dd81f94d806d8889ff1bcb4aed03ce739b8eca1e5e6dc58a4e55c312","target":"graph","created_at":"2026-07-04T15:09:05Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/q-alg/9505019/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This is the fourth part of a series of papers developing a tensor product theory of modules for a vertex operator algebra. In this paper, We establish the associativity of $P(z)$-tensor products for nonzero complex numbers $z$ constructed in Part III of the present series under suitable conditions. The associativity isomorphisms constructed in this paper are analogous to associativity isomorphisms for vector space tensor products in the sense that it relates the tensor products of three elements in three modules taken in different ways. The main new feature is that they are controlled by the d","authors_text":"Yi-Zhi Huang","cross_cats":["hep-th","math.QA"],"headline":"","license":"","primary_cat":"q-alg","submitted_at":"1995-05-17T17:23:59Z","title":"A theory of tensor products for module categories for a vertex operator algebra, IV"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"q-alg/9505019","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:d88397824a7a9690ffff1139f9f5caaa738781c31e9b4b7b9e668838989683c4","target":"record","created_at":"2026-07-04T15:09:05Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"298b46b1aa8162e2264823d5457856b2da6c909909b6c55731b7ea54635ac9ab","cross_cats_sorted":["hep-th","math.QA"],"license":"","primary_cat":"q-alg","submitted_at":"1995-05-17T17:23:59Z","title_canon_sha256":"c9d3644c2413126203064e490af48bbb7232c41b4cc39681d4fe65dd29f98fb0"},"schema_version":"1.0","source":{"id":"q-alg/9505019","kind":"arxiv","version":1}},"canonical_sha256":"d41be8c3d04b8440a36d60703fdd70f7c5284ed66d3d359c1313f2dc5fefdb73","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d41be8c3d04b8440a36d60703fdd70f7c5284ed66d3d359c1313f2dc5fefdb73","first_computed_at":"2026-07-04T15:09:05.752426Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T15:09:05.752426Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2sfaIe0VZb53uHSAJJnk0mr0cHtquYJKFj2zGVWi5e8MkqbOYa5Sz+damOkzq/HlBvd/JHnS8PNUuqb1cp4RAA==","signature_status":"signed_v1","signed_at":"2026-07-04T15:09:05.752803Z","signed_message":"canonical_sha256_bytes"},"source_id":"q-alg/9505019","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d88397824a7a9690ffff1139f9f5caaa738781c31e9b4b7b9e668838989683c4","sha256:f60e0e00dd81f94d806d8889ff1bcb4aed03ce739b8eca1e5e6dc58a4e55c312"],"state_sha256":"89cba5f640d2d852284d1d83e6889d5ced07f251133c050e81d3b385a6692c8d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cd5hhSU6XNLA6vvUDXm5sNVwxJKDC0HcDTNxxNb24pJluaHGHdWbmnA5iL7WbbPUETzyS/cONYQ7g+Vwu91cAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-10T23:32:08.148007Z","bundle_sha256":"ee6453ba1ec683ea7d05e03f34f6dbc55ad105aadb4573a81d833ddffd6a1a15"}}