{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:FGTC2XM5FI373U464PFKYKOFZ2","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":"f7b50915919c2901d0902561f4b7e37a229d0a7c69a2622942b831da931d90e6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2016-10-17T19:50:47Z","title_canon_sha256":"2bdf62793201ee42c2a8b7ab3d3649772ca4f621601e30185b87f3ccf1f403b3"},"schema_version":"1.0","source":{"id":"1610.05291","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.05291","created_at":"2026-05-18T01:02:05Z"},{"alias_kind":"arxiv_version","alias_value":"1610.05291v1","created_at":"2026-05-18T01:02:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.05291","created_at":"2026-05-18T01:02:05Z"},{"alias_kind":"pith_short_12","alias_value":"FGTC2XM5FI37","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"FGTC2XM5FI373U46","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"FGTC2XM5","created_at":"2026-05-18T12:30:15Z"}],"graph_snapshots":[{"event_id":"sha256:d2683c9ee6f3f4f6a7a21df1a6d697daa80c254f9d4a09baaf5855a1647fec5f","target":"graph","created_at":"2026-05-18T01:02:05Z","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"},"paper":{"abstract_excerpt":"Let $V$ be a rational, selfdual, $C_2$-cofinite vertex operator algebra of CFT type, and $G$ a finite automorphism group of $V.$ It is proved that the kernel of the representation of the modular group on twisted conformal blocks associated to $V$ and $G$ is a congruence subgroup. In particular, the $q$-character of each irreducible twisted module is a modular function on the same congruence subgroup. In the case $V$ is the Frenkel-Lepowsky-Meurman's moonshine vertex operator algebra and $G$ is the monster simple group, the generalized McKay-Thompson series associated to any commuting pair in t","authors_text":"Chongying Dong, Li Ren","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2016-10-17T19:50:47Z","title":"Congruence Property in Orbifold Theory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.05291","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:d688e4e46ee0e40ce51895bade69bbab0128b7c726492cbe2a2e941ab81a4d22","target":"record","created_at":"2026-05-18T01:02:05Z","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":"f7b50915919c2901d0902561f4b7e37a229d0a7c69a2622942b831da931d90e6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2016-10-17T19:50:47Z","title_canon_sha256":"2bdf62793201ee42c2a8b7ab3d3649772ca4f621601e30185b87f3ccf1f403b3"},"schema_version":"1.0","source":{"id":"1610.05291","kind":"arxiv","version":1}},"canonical_sha256":"29a62d5d9d2a37fdd39ee3caac29c5ceb7ff00563706e12be47a69f505a5199f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"29a62d5d9d2a37fdd39ee3caac29c5ceb7ff00563706e12be47a69f505a5199f","first_computed_at":"2026-05-18T01:02:05.594972Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:02:05.594972Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2q/vZSb7jGyAOP43jxgVLUaIIuz2hIvJJbH5NO5IPN2VRxA2AKtPMcKjaDcr0qiKaX91YOTdafZmDOXH2D9pDw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:02:05.595652Z","signed_message":"canonical_sha256_bytes"},"source_id":"1610.05291","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d688e4e46ee0e40ce51895bade69bbab0128b7c726492cbe2a2e941ab81a4d22","sha256:d2683c9ee6f3f4f6a7a21df1a6d697daa80c254f9d4a09baaf5855a1647fec5f"],"state_sha256":"c4442ca8935ff78c487c466b1083c042f8da25bd8331f00146ad494be8b9d7d8"}