{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:32JLM5HOMMWJUGUNRJGHFVKVLF","short_pith_number":"pith:32JLM5HO","canonical_record":{"source":{"id":"1611.06611","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2016-11-20T23:31:23Z","cross_cats_sorted":[],"title_canon_sha256":"27007668802cc3880bfe7d108dad2ec77bef4f64ba41d34d4d1dafbe0ee7751a","abstract_canon_sha256":"8dee9a1cbccccfd3d7a112feec738836125f3c93728829131c6e1623777e492e"},"schema_version":"1.0"},"canonical_sha256":"de92b674ee632c9a1a8d8a4c72d5555957412ec9f6b65fa0caf30d2be481689f","source":{"kind":"arxiv","id":"1611.06611","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.06611","created_at":"2026-05-18T00:57:36Z"},{"alias_kind":"arxiv_version","alias_value":"1611.06611v1","created_at":"2026-05-18T00:57:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.06611","created_at":"2026-05-18T00:57:36Z"},{"alias_kind":"pith_short_12","alias_value":"32JLM5HOMMWJ","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_16","alias_value":"32JLM5HOMMWJUGUN","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_8","alias_value":"32JLM5HO","created_at":"2026-05-18T12:29:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:32JLM5HOMMWJUGUNRJGHFVKVLF","target":"record","payload":{"canonical_record":{"source":{"id":"1611.06611","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2016-11-20T23:31:23Z","cross_cats_sorted":[],"title_canon_sha256":"27007668802cc3880bfe7d108dad2ec77bef4f64ba41d34d4d1dafbe0ee7751a","abstract_canon_sha256":"8dee9a1cbccccfd3d7a112feec738836125f3c93728829131c6e1623777e492e"},"schema_version":"1.0"},"canonical_sha256":"de92b674ee632c9a1a8d8a4c72d5555957412ec9f6b65fa0caf30d2be481689f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:57:36.298927Z","signature_b64":"JFttcBbtaGeir/xbDCAcWKOSVxe5vqAKmWTtrSvJhxTQNw4O+4Xkb5nlY6lUH1Ei/SeIJll7CvPPuOia5bseAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"de92b674ee632c9a1a8d8a4c72d5555957412ec9f6b65fa0caf30d2be481689f","last_reissued_at":"2026-05-18T00:57:36.298348Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:57:36.298348Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1611.06611","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":"2026-05-18T00:57:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ODyr9Twa5I0Lf+/v6ZeriexXOYqoayjEYNNbAwfMsJUdJfRXhpfkN90I4/2F4Vlk5lgukjopoVbSxekmpwkmCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T09:51:11.654657Z"},"content_sha256":"95c13ba506868b8e630658ff5c90720907b063f695b91d941bd51d6ad47d0a72","schema_version":"1.0","event_id":"sha256:95c13ba506868b8e630658ff5c90720907b063f695b91d941bd51d6ad47d0a72"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:32JLM5HOMMWJUGUNRJGHFVKVLF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Modular $A_n(V)$ theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.QA","authors_text":"Li Ren","submitted_at":"2016-11-20T23:31:23Z","abstract_excerpt":"A series of associative algebras $A_n(V)$ for a vertex operator algebra $V$ over an arbitrary algebraically closed field and nonnegative integers $n$ are constructed such that there is a one to one correspondence between irreducible $A_n(V)$-modules which are not $A_{n-1}(V)$ modules and irreducible $V$-modules. Moreover, $V$ is rational if and only if $A_n(V)$ is semisimple for all $n.$ In particular, the homogeneous subspaces of any irreducible $V$-module are finite dimensional for rational vertex operator algebra $V.$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.06611","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":"2026-05-18T00:57:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UAmWZ4WZ08TM/Xd7RC26Q46Rr9+xeugLlIdJYd3X8O0eHPVCbaI/G6e2MDSHZCsOxRL2ze09jS6ZjUxYtTUdCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T09:51:11.655218Z"},"content_sha256":"c520607c2f7cfa549ee4e123d4fdf3c148709a263196e9bb37b3840276efc6c3","schema_version":"1.0","event_id":"sha256:c520607c2f7cfa549ee4e123d4fdf3c148709a263196e9bb37b3840276efc6c3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/32JLM5HOMMWJUGUNRJGHFVKVLF/bundle.json","state_url":"https://pith.science/pith/32JLM5HOMMWJUGUNRJGHFVKVLF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/32JLM5HOMMWJUGUNRJGHFVKVLF/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":"2026-06-03T09:51:11Z","links":{"resolver":"https://pith.science/pith/32JLM5HOMMWJUGUNRJGHFVKVLF","bundle":"https://pith.science/pith/32JLM5HOMMWJUGUNRJGHFVKVLF/bundle.json","state":"https://pith.science/pith/32JLM5HOMMWJUGUNRJGHFVKVLF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/32JLM5HOMMWJUGUNRJGHFVKVLF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:32JLM5HOMMWJUGUNRJGHFVKVLF","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":"8dee9a1cbccccfd3d7a112feec738836125f3c93728829131c6e1623777e492e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2016-11-20T23:31:23Z","title_canon_sha256":"27007668802cc3880bfe7d108dad2ec77bef4f64ba41d34d4d1dafbe0ee7751a"},"schema_version":"1.0","source":{"id":"1611.06611","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.06611","created_at":"2026-05-18T00:57:36Z"},{"alias_kind":"arxiv_version","alias_value":"1611.06611v1","created_at":"2026-05-18T00:57:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.06611","created_at":"2026-05-18T00:57:36Z"},{"alias_kind":"pith_short_12","alias_value":"32JLM5HOMMWJ","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_16","alias_value":"32JLM5HOMMWJUGUN","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_8","alias_value":"32JLM5HO","created_at":"2026-05-18T12:29:55Z"}],"graph_snapshots":[{"event_id":"sha256:c520607c2f7cfa549ee4e123d4fdf3c148709a263196e9bb37b3840276efc6c3","target":"graph","created_at":"2026-05-18T00:57:36Z","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":"A series of associative algebras $A_n(V)$ for a vertex operator algebra $V$ over an arbitrary algebraically closed field and nonnegative integers $n$ are constructed such that there is a one to one correspondence between irreducible $A_n(V)$-modules which are not $A_{n-1}(V)$ modules and irreducible $V$-modules. Moreover, $V$ is rational if and only if $A_n(V)$ is semisimple for all $n.$ In particular, the homogeneous subspaces of any irreducible $V$-module are finite dimensional for rational vertex operator algebra $V.$","authors_text":"Li Ren","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2016-11-20T23:31:23Z","title":"Modular $A_n(V)$ theory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.06611","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":"sha256:95c13ba506868b8e630658ff5c90720907b063f695b91d941bd51d6ad47d0a72","target":"record","created_at":"2026-05-18T00:57:36Z","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":"8dee9a1cbccccfd3d7a112feec738836125f3c93728829131c6e1623777e492e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2016-11-20T23:31:23Z","title_canon_sha256":"27007668802cc3880bfe7d108dad2ec77bef4f64ba41d34d4d1dafbe0ee7751a"},"schema_version":"1.0","source":{"id":"1611.06611","kind":"arxiv","version":1}},"canonical_sha256":"de92b674ee632c9a1a8d8a4c72d5555957412ec9f6b65fa0caf30d2be481689f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"de92b674ee632c9a1a8d8a4c72d5555957412ec9f6b65fa0caf30d2be481689f","first_computed_at":"2026-05-18T00:57:36.298348Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:57:36.298348Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JFttcBbtaGeir/xbDCAcWKOSVxe5vqAKmWTtrSvJhxTQNw4O+4Xkb5nlY6lUH1Ei/SeIJll7CvPPuOia5bseAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:57:36.298927Z","signed_message":"canonical_sha256_bytes"},"source_id":"1611.06611","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:95c13ba506868b8e630658ff5c90720907b063f695b91d941bd51d6ad47d0a72","sha256:c520607c2f7cfa549ee4e123d4fdf3c148709a263196e9bb37b3840276efc6c3"],"state_sha256":"388418f34da2b7e2e4fd77622c610bd67050182b1aea0728b921f33e9560ac1c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zm3ZyQd1MuVH/4JBKq8A5N34Xwn99oRB0hcSEoQVyqAEdNy24DR6UsWai9F1+tJ0/n8vQNxNrcw6B0+yrqHWAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T09:51:11.657889Z","bundle_sha256":"87dbe2b0a2a18540ae02dfb1f8a7ee1afdfc429900ce9df9405e3ae2d0c08ef7"}}