{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:UXPPJPCIA3QYMMV5Y3EBXD36X5","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":"1ad83f7f30fb6e94dd383aa61e49c4a581bfdc37a1694ff319b6b122964682a0","cross_cats_sorted":["math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2010-08-31T16:59:53Z","title_canon_sha256":"201db4751f82e711189ca2824fee0258bf02a9694550ff864eafa0a1d7078e41"},"schema_version":"1.0","source":{"id":"1008.5364","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.5364","created_at":"2026-05-18T04:41:14Z"},{"alias_kind":"arxiv_version","alias_value":"1008.5364v2","created_at":"2026-05-18T04:41:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.5364","created_at":"2026-05-18T04:41:14Z"},{"alias_kind":"pith_short_12","alias_value":"UXPPJPCIA3QY","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_16","alias_value":"UXPPJPCIA3QYMMV5","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_8","alias_value":"UXPPJPCI","created_at":"2026-05-18T12:26:15Z"}],"graph_snapshots":[{"event_id":"sha256:04f344d5378bc651ae13477024b559aba1a4f170d76cd70bba6f061ebb221f97","target":"graph","created_at":"2026-05-18T04:41:14Z","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 pairs of graphs (Gamma_k, Gamma'_k), k = 0,1,2,... has been considered as candidates for dual pairs of principal graphs of subfactors of small Jones index above 4 and it has recently been proved that the pair (Gamma_k, Gamma'_k) comes from a subfactor if and only if k = 0 or k =1. We show that nevertheless there exists a unique fusion system compatible with this pair of graphs for all non-negative integers k.","authors_text":"Marta Asaeda, Uffe Haagerup","cross_cats":["math.CT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2010-08-31T16:59:53Z","title":"Fusion rules on a parametrized series of graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.5364","kind":"arxiv","version":2},"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:aa2bf4ee3561776c9b33419d339aa31be32f2fdb75e44c47d11ae565f77b5ac9","target":"record","created_at":"2026-05-18T04:41:14Z","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":"1ad83f7f30fb6e94dd383aa61e49c4a581bfdc37a1694ff319b6b122964682a0","cross_cats_sorted":["math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2010-08-31T16:59:53Z","title_canon_sha256":"201db4751f82e711189ca2824fee0258bf02a9694550ff864eafa0a1d7078e41"},"schema_version":"1.0","source":{"id":"1008.5364","kind":"arxiv","version":2}},"canonical_sha256":"a5def4bc4806e18632bdc6c81b8f7ebf6af2943bc1c7d08b11c0b4c3b902a362","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a5def4bc4806e18632bdc6c81b8f7ebf6af2943bc1c7d08b11c0b4c3b902a362","first_computed_at":"2026-05-18T04:41:14.946789Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:41:14.946789Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fBJuGMO+o3k1gzQlwTpvrqyN44UTfNmBOi52cJCoki6ydZCOFJzm16H/IhEuKZAfZH6f/HW4DyuI3bdpNDXMCA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:41:14.947381Z","signed_message":"canonical_sha256_bytes"},"source_id":"1008.5364","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:aa2bf4ee3561776c9b33419d339aa31be32f2fdb75e44c47d11ae565f77b5ac9","sha256:04f344d5378bc651ae13477024b559aba1a4f170d76cd70bba6f061ebb221f97"],"state_sha256":"8ec5b5b2d5762970c1a188a078c3921d773493792d7eec8c32ada4f437ac388c"}