{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:UGDUIINAMDCVEU2OANNSFYDYFH","short_pith_number":"pith:UGDUIINA","canonical_record":{"source":{"id":"0909.4460","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2009-09-24T14:46:24Z","cross_cats_sorted":[],"title_canon_sha256":"ea705b7f34c7b67ebaa5e25d0c245c5703780e9ead455df0f283109c203573fa","abstract_canon_sha256":"6d5ac82c1cf2aefff0efa93f9e5ea8d1b56014b6ca29407526d67b6e059d58ce"},"schema_version":"1.0"},"canonical_sha256":"a1874421a060c552534e035b22e07829e117aaa36da44a59391a2f4e1dbc1119","source":{"kind":"arxiv","id":"0909.4460","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0909.4460","created_at":"2026-05-18T04:27:36Z"},{"alias_kind":"arxiv_version","alias_value":"0909.4460v1","created_at":"2026-05-18T04:27:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0909.4460","created_at":"2026-05-18T04:27:36Z"},{"alias_kind":"pith_short_12","alias_value":"UGDUIINAMDCV","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_16","alias_value":"UGDUIINAMDCVEU2O","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_8","alias_value":"UGDUIINA","created_at":"2026-05-18T12:26:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:UGDUIINAMDCVEU2OANNSFYDYFH","target":"record","payload":{"canonical_record":{"source":{"id":"0909.4460","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2009-09-24T14:46:24Z","cross_cats_sorted":[],"title_canon_sha256":"ea705b7f34c7b67ebaa5e25d0c245c5703780e9ead455df0f283109c203573fa","abstract_canon_sha256":"6d5ac82c1cf2aefff0efa93f9e5ea8d1b56014b6ca29407526d67b6e059d58ce"},"schema_version":"1.0"},"canonical_sha256":"a1874421a060c552534e035b22e07829e117aaa36da44a59391a2f4e1dbc1119","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:27:36.376171Z","signature_b64":"EnFgqyeYot4NckJUfYX5FiV8oSciytcE15uc3jGyRmI6+eebEfZvJlQIXVsV0Y9V7tKP7GqprzIF3Zh5jWthDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a1874421a060c552534e035b22e07829e117aaa36da44a59391a2f4e1dbc1119","last_reissued_at":"2026-05-18T04:27:36.375544Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:27:36.375544Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0909.4460","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":"2026-05-18T04:27:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QYEwGjyJKWFEHCl4wJVTvfIg7HfF+sKbSaPUJUWfnodXzaSVLf4RlfAs3w4lcSbZjo98C0LJ5BJzGAB2Z+eNCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T14:20:25.770124Z"},"content_sha256":"42582c28b552d826b1392a17221c417d884caf53640d962309b7bd0f3e55867f","schema_version":"1.0","event_id":"sha256:42582c28b552d826b1392a17221c417d884caf53640d962309b7bd0f3e55867f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:UGDUIINAMDCVEU2OANNSFYDYFH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Vertex Operators and Modular Forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.QA","authors_text":"Geoffrey Mason, Michael P. Tuite","submitted_at":"2009-09-24T14:46:24Z","abstract_excerpt":"The leitmotif of these Notes is the idea of a vertex operator algebra (VOA) and the relationship between VOAs and elliptic functions and modular forms. This is to some extent analogous to the relationship between a finite group and its irreducible characters; the algebraic structure determines a set of numerical invariants, and arithmetic properties of the invariants provides feedback in the form of restrictions on the algebraic structure. One of the main points of these Notes is to explain how this works, and to give some reasonably interesting examples."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.4460","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":""},"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":"2026-05-18T04:27:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AjvU0JSSIo0yyYFqowKIUQ0Ng+CIMFdBwu9ix8UNtLE8F0JgnBx62SkrViOIjyP9Uq+PcfpnspNf3uk1zdasDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T14:20:25.770494Z"},"content_sha256":"61d31e549fc25b984f6016772e8c3ad0f40ab7d2541ef4465f92f7367bfb4b21","schema_version":"1.0","event_id":"sha256:61d31e549fc25b984f6016772e8c3ad0f40ab7d2541ef4465f92f7367bfb4b21"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UGDUIINAMDCVEU2OANNSFYDYFH/bundle.json","state_url":"https://pith.science/pith/UGDUIINAMDCVEU2OANNSFYDYFH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UGDUIINAMDCVEU2OANNSFYDYFH/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":"2026-06-03T14:20:25Z","links":{"resolver":"https://pith.science/pith/UGDUIINAMDCVEU2OANNSFYDYFH","bundle":"https://pith.science/pith/UGDUIINAMDCVEU2OANNSFYDYFH/bundle.json","state":"https://pith.science/pith/UGDUIINAMDCVEU2OANNSFYDYFH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UGDUIINAMDCVEU2OANNSFYDYFH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:UGDUIINAMDCVEU2OANNSFYDYFH","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":"6d5ac82c1cf2aefff0efa93f9e5ea8d1b56014b6ca29407526d67b6e059d58ce","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2009-09-24T14:46:24Z","title_canon_sha256":"ea705b7f34c7b67ebaa5e25d0c245c5703780e9ead455df0f283109c203573fa"},"schema_version":"1.0","source":{"id":"0909.4460","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0909.4460","created_at":"2026-05-18T04:27:36Z"},{"alias_kind":"arxiv_version","alias_value":"0909.4460v1","created_at":"2026-05-18T04:27:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0909.4460","created_at":"2026-05-18T04:27:36Z"},{"alias_kind":"pith_short_12","alias_value":"UGDUIINAMDCV","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_16","alias_value":"UGDUIINAMDCVEU2O","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_8","alias_value":"UGDUIINA","created_at":"2026-05-18T12:26:02Z"}],"graph_snapshots":[{"event_id":"sha256:61d31e549fc25b984f6016772e8c3ad0f40ab7d2541ef4465f92f7367bfb4b21","target":"graph","created_at":"2026-05-18T04:27:36Z","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"},"paper":{"abstract_excerpt":"The leitmotif of these Notes is the idea of a vertex operator algebra (VOA) and the relationship between VOAs and elliptic functions and modular forms. This is to some extent analogous to the relationship between a finite group and its irreducible characters; the algebraic structure determines a set of numerical invariants, and arithmetic properties of the invariants provides feedback in the form of restrictions on the algebraic structure. One of the main points of these Notes is to explain how this works, and to give some reasonably interesting examples.","authors_text":"Geoffrey Mason, Michael P. Tuite","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2009-09-24T14:46:24Z","title":"Vertex Operators and Modular Forms"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.4460","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":"sha256:42582c28b552d826b1392a17221c417d884caf53640d962309b7bd0f3e55867f","target":"record","created_at":"2026-05-18T04:27:36Z","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":"6d5ac82c1cf2aefff0efa93f9e5ea8d1b56014b6ca29407526d67b6e059d58ce","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2009-09-24T14:46:24Z","title_canon_sha256":"ea705b7f34c7b67ebaa5e25d0c245c5703780e9ead455df0f283109c203573fa"},"schema_version":"1.0","source":{"id":"0909.4460","kind":"arxiv","version":1}},"canonical_sha256":"a1874421a060c552534e035b22e07829e117aaa36da44a59391a2f4e1dbc1119","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a1874421a060c552534e035b22e07829e117aaa36da44a59391a2f4e1dbc1119","first_computed_at":"2026-05-18T04:27:36.375544Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:27:36.375544Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EnFgqyeYot4NckJUfYX5FiV8oSciytcE15uc3jGyRmI6+eebEfZvJlQIXVsV0Y9V7tKP7GqprzIF3Zh5jWthDA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:27:36.376171Z","signed_message":"canonical_sha256_bytes"},"source_id":"0909.4460","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:42582c28b552d826b1392a17221c417d884caf53640d962309b7bd0f3e55867f","sha256:61d31e549fc25b984f6016772e8c3ad0f40ab7d2541ef4465f92f7367bfb4b21"],"state_sha256":"4264a194a77d3724323243e95c753f0b17b9d4cfc3f8a4493869612b57ac27b0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3z57xfnRRA+K3LE17NEjOMFyzZdsWXwFlzmKAi4tAg7olPHz8qDoI+MlWvkTSMuJ6MJ/g5YhwUuteWUBNF+0Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T14:20:25.772425Z","bundle_sha256":"01300bbdda7de84615107836f05bd2e529452c2ee2e04616fcc36b9bab20420e"}}