{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:TNPGETQT6NHZYMWXC5KB4YERK6","short_pith_number":"pith:TNPGETQT","canonical_record":{"source":{"id":"1110.1199","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-10-06T09:53:00Z","cross_cats_sorted":[],"title_canon_sha256":"2b5bb297d8414f6907a0d7b6c89ad01e2674ebc4e8fa2f8cc4a7668095db287c","abstract_canon_sha256":"803c0a79b05daa2749a029d4116a847765b426055121a8bc93d18d008b74fb87"},"schema_version":"1.0"},"canonical_sha256":"9b5e624e13f34f9c32d717541e609157ae77e84a518c85a7b14b61af0e33c654","source":{"kind":"arxiv","id":"1110.1199","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.1199","created_at":"2026-05-18T03:26:10Z"},{"alias_kind":"arxiv_version","alias_value":"1110.1199v4","created_at":"2026-05-18T03:26:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.1199","created_at":"2026-05-18T03:26:10Z"},{"alias_kind":"pith_short_12","alias_value":"TNPGETQT6NHZ","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_16","alias_value":"TNPGETQT6NHZYMWX","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_8","alias_value":"TNPGETQT","created_at":"2026-05-18T12:26:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:TNPGETQT6NHZYMWXC5KB4YERK6","target":"record","payload":{"canonical_record":{"source":{"id":"1110.1199","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-10-06T09:53:00Z","cross_cats_sorted":[],"title_canon_sha256":"2b5bb297d8414f6907a0d7b6c89ad01e2674ebc4e8fa2f8cc4a7668095db287c","abstract_canon_sha256":"803c0a79b05daa2749a029d4116a847765b426055121a8bc93d18d008b74fb87"},"schema_version":"1.0"},"canonical_sha256":"9b5e624e13f34f9c32d717541e609157ae77e84a518c85a7b14b61af0e33c654","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:26:10.476496Z","signature_b64":"Xn1IG36wdvzbBMwDdpBY6SfeR8frN2PgDmIHu+A+M5F3Qyesxuma+vqQNh8aO8Sl2ID1Qkw+O2iwXwuMUi96BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9b5e624e13f34f9c32d717541e609157ae77e84a518c85a7b14b61af0e33c654","last_reissued_at":"2026-05-18T03:26:10.475718Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:26:10.475718Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1110.1199","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:26:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AMir+/FF0LsY3sjUKGL1/qq85DEAHP3lIujKgUv8opSlZe3CN2kYJRnkwleZB3wRBOfIITxpjHFTaY/Q/jXxBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T11:52:21.452786Z"},"content_sha256":"2f2ac3511b581e75c4ad99d127b01f571afa0fe722f93892cb565969830904a2","schema_version":"1.0","event_id":"sha256:2f2ac3511b581e75c4ad99d127b01f571afa0fe722f93892cb565969830904a2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:TNPGETQT6NHZYMWXC5KB4YERK6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Factorial cluster algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Bernard Leclerc, Christof Gei{\\ss}, Jan Schr\\\"oer","submitted_at":"2011-10-06T09:53:00Z","abstract_excerpt":"We show that cluster algebras do not contain non-trivial units and that all cluster variables are irreducible elements. Both statements follow from Fomin and Zelevinsky's Laurent phenomenon. As an application we give a criterion for a cluster algebra to be a factorial algebra. This can be used to construct cluster algebras, which are isomorphic to polynomial rings. We also study various kinds of upper bounds for cluster algebras, and we prove that factorial cluster algebras coincide with their upper bounds."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.1199","kind":"arxiv","version":4},"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":"2026-05-18T03:26:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HcT68rNg3wjPkU88pvRTnVzEuyaelZ5UtLPJQI0jKNHg+PVHEfyoGpS8SWbtUSO1kBM46e6rnu9GLORzsB71DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T11:52:21.453367Z"},"content_sha256":"a3f1c9893916e2de5adc91c0d522bf3318b495010d23ef13814db9f0037957e9","schema_version":"1.0","event_id":"sha256:a3f1c9893916e2de5adc91c0d522bf3318b495010d23ef13814db9f0037957e9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TNPGETQT6NHZYMWXC5KB4YERK6/bundle.json","state_url":"https://pith.science/pith/TNPGETQT6NHZYMWXC5KB4YERK6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TNPGETQT6NHZYMWXC5KB4YERK6/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":"2026-08-20T11:52:21Z","links":{"resolver":"https://pith.science/pith/TNPGETQT6NHZYMWXC5KB4YERK6","bundle":"https://pith.science/pith/TNPGETQT6NHZYMWXC5KB4YERK6/bundle.json","state":"https://pith.science/pith/TNPGETQT6NHZYMWXC5KB4YERK6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TNPGETQT6NHZYMWXC5KB4YERK6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:TNPGETQT6NHZYMWXC5KB4YERK6","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":"803c0a79b05daa2749a029d4116a847765b426055121a8bc93d18d008b74fb87","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-10-06T09:53:00Z","title_canon_sha256":"2b5bb297d8414f6907a0d7b6c89ad01e2674ebc4e8fa2f8cc4a7668095db287c"},"schema_version":"1.0","source":{"id":"1110.1199","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.1199","created_at":"2026-05-18T03:26:10Z"},{"alias_kind":"arxiv_version","alias_value":"1110.1199v4","created_at":"2026-05-18T03:26:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.1199","created_at":"2026-05-18T03:26:10Z"},{"alias_kind":"pith_short_12","alias_value":"TNPGETQT6NHZ","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_16","alias_value":"TNPGETQT6NHZYMWX","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_8","alias_value":"TNPGETQT","created_at":"2026-05-18T12:26:42Z"}],"graph_snapshots":[{"event_id":"sha256:a3f1c9893916e2de5adc91c0d522bf3318b495010d23ef13814db9f0037957e9","target":"graph","created_at":"2026-05-18T03:26:10Z","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":"We show that cluster algebras do not contain non-trivial units and that all cluster variables are irreducible elements. Both statements follow from Fomin and Zelevinsky's Laurent phenomenon. As an application we give a criterion for a cluster algebra to be a factorial algebra. This can be used to construct cluster algebras, which are isomorphic to polynomial rings. We also study various kinds of upper bounds for cluster algebras, and we prove that factorial cluster algebras coincide with their upper bounds.","authors_text":"Bernard Leclerc, Christof Gei{\\ss}, Jan Schr\\\"oer","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-10-06T09:53:00Z","title":"Factorial cluster algebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.1199","kind":"arxiv","version":4},"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":"sha256:2f2ac3511b581e75c4ad99d127b01f571afa0fe722f93892cb565969830904a2","target":"record","created_at":"2026-05-18T03:26:10Z","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":"803c0a79b05daa2749a029d4116a847765b426055121a8bc93d18d008b74fb87","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-10-06T09:53:00Z","title_canon_sha256":"2b5bb297d8414f6907a0d7b6c89ad01e2674ebc4e8fa2f8cc4a7668095db287c"},"schema_version":"1.0","source":{"id":"1110.1199","kind":"arxiv","version":4}},"canonical_sha256":"9b5e624e13f34f9c32d717541e609157ae77e84a518c85a7b14b61af0e33c654","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9b5e624e13f34f9c32d717541e609157ae77e84a518c85a7b14b61af0e33c654","first_computed_at":"2026-05-18T03:26:10.475718Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:26:10.475718Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Xn1IG36wdvzbBMwDdpBY6SfeR8frN2PgDmIHu+A+M5F3Qyesxuma+vqQNh8aO8Sl2ID1Qkw+O2iwXwuMUi96BA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:26:10.476496Z","signed_message":"canonical_sha256_bytes"},"source_id":"1110.1199","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2f2ac3511b581e75c4ad99d127b01f571afa0fe722f93892cb565969830904a2","sha256:a3f1c9893916e2de5adc91c0d522bf3318b495010d23ef13814db9f0037957e9"],"state_sha256":"1e10be5d1bbc968457da9a75e24d827baedd1ef0da266962b2b2ece7cf7bb4b0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4KwVt6h2blNeBlDK8fLMZ9PGdAG5wvbWXJPA05mL1xz6GyKQDC/UOWKk76/xQIPWCmtqFIH/WoFvILarEwXRBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-20T11:52:21.457368Z","bundle_sha256":"49d1973ae8078d3b34c4cb0346804e0eb7779178727c16f51ee96173115e89a8"}}