{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:ZAAQU3F7ONBYQLD2LCFJ53N7ND","short_pith_number":"pith:ZAAQU3F7","canonical_record":{"source":{"id":"1909.01693","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2019-09-04T11:03:05Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"f38a126b1c4ddf8633e1558fe4100a6f5876fd75623b9289a9b2f736b3214c8f","abstract_canon_sha256":"db08526c4d9dfedac1abc1bf6ddb63284f3a3d51607cb060a31dfcca4ae5c45f"},"schema_version":"1.0"},"canonical_sha256":"c8010a6cbf7343882c7a588a9eedbf68d6d3495bdd7ee3ae71e75a2a6cb74811","source":{"kind":"arxiv","id":"1909.01693","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1909.01693","created_at":"2026-07-05T03:59:20Z"},{"alias_kind":"arxiv_version","alias_value":"1909.01693v3","created_at":"2026-07-05T03:59:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1909.01693","created_at":"2026-07-05T03:59:20Z"},{"alias_kind":"pith_short_12","alias_value":"ZAAQU3F7ONBY","created_at":"2026-07-05T03:59:20Z"},{"alias_kind":"pith_short_16","alias_value":"ZAAQU3F7ONBYQLD2","created_at":"2026-07-05T03:59:20Z"},{"alias_kind":"pith_short_8","alias_value":"ZAAQU3F7","created_at":"2026-07-05T03:59:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:ZAAQU3F7ONBYQLD2LCFJ53N7ND","target":"record","payload":{"canonical_record":{"source":{"id":"1909.01693","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2019-09-04T11:03:05Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"f38a126b1c4ddf8633e1558fe4100a6f5876fd75623b9289a9b2f736b3214c8f","abstract_canon_sha256":"db08526c4d9dfedac1abc1bf6ddb63284f3a3d51607cb060a31dfcca4ae5c45f"},"schema_version":"1.0"},"canonical_sha256":"c8010a6cbf7343882c7a588a9eedbf68d6d3495bdd7ee3ae71e75a2a6cb74811","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:59:20.152247Z","signature_b64":"3dbrWEzuP3MxB0GzEc0Z9J7zx//Gqef+jWJOxF4cp00Sk7s93lCwDCXaa7RNXDCg6N9SsHJ8xVOi57JMldLABg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c8010a6cbf7343882c7a588a9eedbf68d6d3495bdd7ee3ae71e75a2a6cb74811","last_reissued_at":"2026-07-05T03:59:20.151799Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:59:20.151799Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1909.01693","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T03:59:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jGhLPK4SsNEDXZiU/6qz2R10d+wVgzZmPWA/agA/c3wl2dU8NBur/loRyqtMF0RvbHHh4+xxw3Q8vI0g9OsDBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-15T18:40:04.158697Z"},"content_sha256":"de67ebe319e8c418f2d440c66f82b9d086fff0dd8cd64c7c006133728ba7f07b","schema_version":"1.0","event_id":"sha256:de67ebe319e8c418f2d440c66f82b9d086fff0dd8cd64c7c006133728ba7f07b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:ZAAQU3F7ONBYQLD2LCFJ53N7ND","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Frobenius-Perron Dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.AG","authors_text":"Changzheng Li, Chi Zhang, Mingzhi Yang, Ryan M. Shifler","submitted_at":"2019-09-04T11:03:05Z","abstract_excerpt":"We propose a notion of Frobenius-Perron dimension for certain free $\\mathbb{Z}$-modules of infinite rank and compute it for the $\\mathbb{Z}$-modules of finite dimensional complex representations of unitary groups with nonnegative dominant weights. We also provide a lower bound for the Frobenius-Perron dimension of Schubert classes in the quantum cohomology of complex Grassmannians."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1909.01693","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1909.01693/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T03:59:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jnnlKxZQkln3uwV96dljWF+IxzFUFUDytwor1isxL+V8WU2wZFgWu56TCZQiVHrHNcfe3zzqNihcfGxAuCOsCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-15T18:40:04.159187Z"},"content_sha256":"f5d1fcac6c8ec927ea5eca3a328bba24371bf2a6c388fc4fb985cc6e28482ea5","schema_version":"1.0","event_id":"sha256:f5d1fcac6c8ec927ea5eca3a328bba24371bf2a6c388fc4fb985cc6e28482ea5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZAAQU3F7ONBYQLD2LCFJ53N7ND/bundle.json","state_url":"https://pith.science/pith/ZAAQU3F7ONBYQLD2LCFJ53N7ND/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZAAQU3F7ONBYQLD2LCFJ53N7ND/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":"2026-08-15T18:40:04Z","links":{"resolver":"https://pith.science/pith/ZAAQU3F7ONBYQLD2LCFJ53N7ND","bundle":"https://pith.science/pith/ZAAQU3F7ONBYQLD2LCFJ53N7ND/bundle.json","state":"https://pith.science/pith/ZAAQU3F7ONBYQLD2LCFJ53N7ND/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZAAQU3F7ONBYQLD2LCFJ53N7ND/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:ZAAQU3F7ONBYQLD2LCFJ53N7ND","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":"db08526c4d9dfedac1abc1bf6ddb63284f3a3d51607cb060a31dfcca4ae5c45f","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2019-09-04T11:03:05Z","title_canon_sha256":"f38a126b1c4ddf8633e1558fe4100a6f5876fd75623b9289a9b2f736b3214c8f"},"schema_version":"1.0","source":{"id":"1909.01693","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1909.01693","created_at":"2026-07-05T03:59:20Z"},{"alias_kind":"arxiv_version","alias_value":"1909.01693v3","created_at":"2026-07-05T03:59:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1909.01693","created_at":"2026-07-05T03:59:20Z"},{"alias_kind":"pith_short_12","alias_value":"ZAAQU3F7ONBY","created_at":"2026-07-05T03:59:20Z"},{"alias_kind":"pith_short_16","alias_value":"ZAAQU3F7ONBYQLD2","created_at":"2026-07-05T03:59:20Z"},{"alias_kind":"pith_short_8","alias_value":"ZAAQU3F7","created_at":"2026-07-05T03:59:20Z"}],"graph_snapshots":[{"event_id":"sha256:f5d1fcac6c8ec927ea5eca3a328bba24371bf2a6c388fc4fb985cc6e28482ea5","target":"graph","created_at":"2026-07-05T03:59:20Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1909.01693/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We propose a notion of Frobenius-Perron dimension for certain free $\\mathbb{Z}$-modules of infinite rank and compute it for the $\\mathbb{Z}$-modules of finite dimensional complex representations of unitary groups with nonnegative dominant weights. We also provide a lower bound for the Frobenius-Perron dimension of Schubert classes in the quantum cohomology of complex Grassmannians.","authors_text":"Changzheng Li, Chi Zhang, Mingzhi Yang, Ryan M. Shifler","cross_cats":["math.RA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2019-09-04T11:03:05Z","title":"On Frobenius-Perron Dimension"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1909.01693","kind":"arxiv","version":3},"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:de67ebe319e8c418f2d440c66f82b9d086fff0dd8cd64c7c006133728ba7f07b","target":"record","created_at":"2026-07-05T03:59:20Z","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":"db08526c4d9dfedac1abc1bf6ddb63284f3a3d51607cb060a31dfcca4ae5c45f","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2019-09-04T11:03:05Z","title_canon_sha256":"f38a126b1c4ddf8633e1558fe4100a6f5876fd75623b9289a9b2f736b3214c8f"},"schema_version":"1.0","source":{"id":"1909.01693","kind":"arxiv","version":3}},"canonical_sha256":"c8010a6cbf7343882c7a588a9eedbf68d6d3495bdd7ee3ae71e75a2a6cb74811","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c8010a6cbf7343882c7a588a9eedbf68d6d3495bdd7ee3ae71e75a2a6cb74811","first_computed_at":"2026-07-05T03:59:20.151799Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:59:20.151799Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3dbrWEzuP3MxB0GzEc0Z9J7zx//Gqef+jWJOxF4cp00Sk7s93lCwDCXaa7RNXDCg6N9SsHJ8xVOi57JMldLABg==","signature_status":"signed_v1","signed_at":"2026-07-05T03:59:20.152247Z","signed_message":"canonical_sha256_bytes"},"source_id":"1909.01693","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:de67ebe319e8c418f2d440c66f82b9d086fff0dd8cd64c7c006133728ba7f07b","sha256:f5d1fcac6c8ec927ea5eca3a328bba24371bf2a6c388fc4fb985cc6e28482ea5"],"state_sha256":"fd96587dc85c49f7f85b90b5c6f69339813e038fec74dfa0a4b769659eff2a7b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bcaBVwEkn4oDAd5d46ymNKVNoLg75iAKgsN98So4liLCQGYD49MoKo0QGWB4sO1Wdh6ygPAf+etIru2KbUP1Dg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-15T18:40:04.164208Z","bundle_sha256":"fe7a75ab5eda3a8c73a87c922ec3133d84f0853d2640cfe5714d5107711ab59b"}}