{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:BGDISHSPQSBYUHUS6LYXII53SN","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":"032397160188a2dd743b60465e1299a4a5170b604b4c96dbb6d96cd54d5f5926","cross_cats_sorted":["math.RT"],"license":"","primary_cat":"math.RA","submitted_at":"2006-12-08T11:53:03Z","title_canon_sha256":"1e6da950fcf11410aceba0407b1a8dfa837001bc17b7db5f62d72d61785c2ee5"},"schema_version":"1.0","source":{"id":"math/0612213","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0612213","created_at":"2026-05-18T04:08:53Z"},{"alias_kind":"arxiv_version","alias_value":"math/0612213v2","created_at":"2026-05-18T04:08:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0612213","created_at":"2026-05-18T04:08:53Z"},{"alias_kind":"pith_short_12","alias_value":"BGDISHSPQSBY","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"BGDISHSPQSBYUHUS","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"BGDISHSP","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:7def67bb52d00b6832ece162e1254a53be574e4bbbb463e21af9bc34fecd9319","target":"graph","created_at":"2026-05-18T04:08:53Z","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":"Acyclic cluster algebras have an interpretation in terms of tilting objects in a Calabi-Yau category defined by some hereditary algebra. For a given quiver $Q$ it is thus desirable to decide if the cluster algebra defined by $Q$ is acyclic. We call $Q$ cluster-acyclic in this case, otherwise cluster-cyclic.\n  In this note we classify the cluster-cyclic quivers with three vertices using a Diophantine equation studied by Markov.","authors_text":"Andre Beineke, Lutz Hille, Thomas Br\\\"ustle","cross_cats":["math.RT"],"headline":"","license":"","primary_cat":"math.RA","submitted_at":"2006-12-08T11:53:03Z","title":"Cluster-Cyclic Quivers with three Vertices and the Markov Equation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0612213","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":"sha256:04c9118eee7d732b0faf189764a3ef3e79d753a4c482b6b93426b2c7b36d70c9","target":"record","created_at":"2026-05-18T04:08:53Z","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":"032397160188a2dd743b60465e1299a4a5170b604b4c96dbb6d96cd54d5f5926","cross_cats_sorted":["math.RT"],"license":"","primary_cat":"math.RA","submitted_at":"2006-12-08T11:53:03Z","title_canon_sha256":"1e6da950fcf11410aceba0407b1a8dfa837001bc17b7db5f62d72d61785c2ee5"},"schema_version":"1.0","source":{"id":"math/0612213","kind":"arxiv","version":2}},"canonical_sha256":"0986891e4f84838a1e92f2f17423bb9374178103d54fadd0983c00b8bdd6130a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0986891e4f84838a1e92f2f17423bb9374178103d54fadd0983c00b8bdd6130a","first_computed_at":"2026-05-18T04:08:53.092759Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:08:53.092759Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"N6FnleWleFZJ+1fty2FeXn6R+4blocR9RFQQno6tpfiWyNqCv6ju+HwW079u/GY1m/2jvDGu/3DTG92QCUR2BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:08:53.093196Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0612213","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:04c9118eee7d732b0faf189764a3ef3e79d753a4c482b6b93426b2c7b36d70c9","sha256:7def67bb52d00b6832ece162e1254a53be574e4bbbb463e21af9bc34fecd9319"],"state_sha256":"9fdc4452b48f577a6c2b5d7828df394011fcd1aab84485567d28576fb493ba42"}