{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:4ZNJLHNPAC4F6TWDC57NEMPLO5","short_pith_number":"pith:4ZNJLHNP","canonical_record":{"source":{"id":"1110.2382","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2011-10-11T14:07:38Z","cross_cats_sorted":["cs.DM","math.NT"],"title_canon_sha256":"390b5e383e8d80dec19d4efaf00fd246c5ed876da61cfcbc4ffa88da2c764580","abstract_canon_sha256":"7404443a657a1bd3e58d26c8ec18b8ace9836b4017fb9defb56a9b0d201c2010"},"schema_version":"1.0"},"canonical_sha256":"e65a959daf00b85f4ec3177ed231eb775339148fef8c536bbeb4e6b20ef18a86","source":{"kind":"arxiv","id":"1110.2382","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.2382","created_at":"2026-05-18T01:34:31Z"},{"alias_kind":"arxiv_version","alias_value":"1110.2382v4","created_at":"2026-05-18T01:34:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.2382","created_at":"2026-05-18T01:34:31Z"},{"alias_kind":"pith_short_12","alias_value":"4ZNJLHNPAC4F","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_16","alias_value":"4ZNJLHNPAC4F6TWD","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_8","alias_value":"4ZNJLHNP","created_at":"2026-05-18T12:26:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:4ZNJLHNPAC4F6TWDC57NEMPLO5","target":"record","payload":{"canonical_record":{"source":{"id":"1110.2382","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2011-10-11T14:07:38Z","cross_cats_sorted":["cs.DM","math.NT"],"title_canon_sha256":"390b5e383e8d80dec19d4efaf00fd246c5ed876da61cfcbc4ffa88da2c764580","abstract_canon_sha256":"7404443a657a1bd3e58d26c8ec18b8ace9836b4017fb9defb56a9b0d201c2010"},"schema_version":"1.0"},"canonical_sha256":"e65a959daf00b85f4ec3177ed231eb775339148fef8c536bbeb4e6b20ef18a86","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:34:31.063747Z","signature_b64":"mAvtKWPO+LkL8GeasU3TXv7nZQvYRBeRhTBxWheWpf0sBcZZUT17sDyddRi1rNE+4VI7bDPH0p8yqsBBCGJ2Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e65a959daf00b85f4ec3177ed231eb775339148fef8c536bbeb4e6b20ef18a86","last_reissued_at":"2026-05-18T01:34:31.063326Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:34:31.063326Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1110.2382","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":"2026-05-18T01:34:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pjEBAaik9lSffVn+MAzFJeBGgDSUbMlfLgoVgkRQdAxfJOXYh1LlykNDxfC6dCrdCAHA+FoYcl+DsjA1tydCBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T23:31:01.812985Z"},"content_sha256":"9e1b7469e98174909cf390b294e79351e7cba3ec2809b77bb4cf80da67bf7506","schema_version":"1.0","event_id":"sha256:9e1b7469e98174909cf390b294e79351e7cba3ec2809b77bb4cf80da67bf7506"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:4ZNJLHNPAC4F6TWDC57NEMPLO5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Automatic sets of rational numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.NT"],"primary_cat":"cs.FL","authors_text":"Eric Rowland, Jeffrey Shallit","submitted_at":"2011-10-11T14:07:38Z","abstract_excerpt":"The notion of a k-automatic set of integers is well-studied. We develop a new notion - the k-automatic set of rational numbers - and prove basic properties of these sets, including closure properties and decidability."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.2382","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":"2026-05-18T01:34:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vhDaapHkiIcnuAGq0WptIqDgZ/s8xGvUmxO0Xf+Ra52f+jfX04M+emO7nNfWar4wlStiHbp5T95f1TApME/EDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T23:31:01.813665Z"},"content_sha256":"aff17d3fb8fc8b47f546b127463db4166ce15c11727a657c1fe8bd12f5ae523b","schema_version":"1.0","event_id":"sha256:aff17d3fb8fc8b47f546b127463db4166ce15c11727a657c1fe8bd12f5ae523b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4ZNJLHNPAC4F6TWDC57NEMPLO5/bundle.json","state_url":"https://pith.science/pith/4ZNJLHNPAC4F6TWDC57NEMPLO5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4ZNJLHNPAC4F6TWDC57NEMPLO5/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":"2026-06-04T23:31:01Z","links":{"resolver":"https://pith.science/pith/4ZNJLHNPAC4F6TWDC57NEMPLO5","bundle":"https://pith.science/pith/4ZNJLHNPAC4F6TWDC57NEMPLO5/bundle.json","state":"https://pith.science/pith/4ZNJLHNPAC4F6TWDC57NEMPLO5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4ZNJLHNPAC4F6TWDC57NEMPLO5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:4ZNJLHNPAC4F6TWDC57NEMPLO5","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":"7404443a657a1bd3e58d26c8ec18b8ace9836b4017fb9defb56a9b0d201c2010","cross_cats_sorted":["cs.DM","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2011-10-11T14:07:38Z","title_canon_sha256":"390b5e383e8d80dec19d4efaf00fd246c5ed876da61cfcbc4ffa88da2c764580"},"schema_version":"1.0","source":{"id":"1110.2382","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.2382","created_at":"2026-05-18T01:34:31Z"},{"alias_kind":"arxiv_version","alias_value":"1110.2382v4","created_at":"2026-05-18T01:34:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.2382","created_at":"2026-05-18T01:34:31Z"},{"alias_kind":"pith_short_12","alias_value":"4ZNJLHNPAC4F","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_16","alias_value":"4ZNJLHNPAC4F6TWD","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_8","alias_value":"4ZNJLHNP","created_at":"2026-05-18T12:26:20Z"}],"graph_snapshots":[{"event_id":"sha256:aff17d3fb8fc8b47f546b127463db4166ce15c11727a657c1fe8bd12f5ae523b","target":"graph","created_at":"2026-05-18T01:34:31Z","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":"The notion of a k-automatic set of integers is well-studied. We develop a new notion - the k-automatic set of rational numbers - and prove basic properties of these sets, including closure properties and decidability.","authors_text":"Eric Rowland, Jeffrey Shallit","cross_cats":["cs.DM","math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2011-10-11T14:07:38Z","title":"Automatic sets of rational numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.2382","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":"sha256:9e1b7469e98174909cf390b294e79351e7cba3ec2809b77bb4cf80da67bf7506","target":"record","created_at":"2026-05-18T01:34:31Z","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":"7404443a657a1bd3e58d26c8ec18b8ace9836b4017fb9defb56a9b0d201c2010","cross_cats_sorted":["cs.DM","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2011-10-11T14:07:38Z","title_canon_sha256":"390b5e383e8d80dec19d4efaf00fd246c5ed876da61cfcbc4ffa88da2c764580"},"schema_version":"1.0","source":{"id":"1110.2382","kind":"arxiv","version":4}},"canonical_sha256":"e65a959daf00b85f4ec3177ed231eb775339148fef8c536bbeb4e6b20ef18a86","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e65a959daf00b85f4ec3177ed231eb775339148fef8c536bbeb4e6b20ef18a86","first_computed_at":"2026-05-18T01:34:31.063326Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:34:31.063326Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mAvtKWPO+LkL8GeasU3TXv7nZQvYRBeRhTBxWheWpf0sBcZZUT17sDyddRi1rNE+4VI7bDPH0p8yqsBBCGJ2Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:34:31.063747Z","signed_message":"canonical_sha256_bytes"},"source_id":"1110.2382","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9e1b7469e98174909cf390b294e79351e7cba3ec2809b77bb4cf80da67bf7506","sha256:aff17d3fb8fc8b47f546b127463db4166ce15c11727a657c1fe8bd12f5ae523b"],"state_sha256":"7543b45fcde7fb12a536d51963b041d8d1cdee24ebcf6006d5961bcd3643e846"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Pe0vzYD4toPmKRI72ZnBhG8+jhpwIqo3BWgJItKWClMkczAsXeOIJfDe91xIHkrlC3NVfNLvP+xBA2EPYK/VCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T23:31:01.817307Z","bundle_sha256":"d41b3f51cb497584abb26f8426aafb0d171e238db76b8c503a1209e48b98413a"}}