{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:4EAWMCN6UKPXWKKY4I7RBLBIJN","short_pith_number":"pith:4EAWMCN6","canonical_record":{"source":{"id":"1104.2303","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2011-04-12T18:49:23Z","cross_cats_sorted":["cs.DM","math.NT"],"title_canon_sha256":"2d9cacf599b6a69feaa009462af81fd5144df32ff1dcc6bbb848c6840166a074","abstract_canon_sha256":"0c689197272f4aae93047a8efc245363cd14f5265dd5ab90491163a867e6b798"},"schema_version":"1.0"},"canonical_sha256":"e1016609bea29f7b2958e23f10ac284b59791de815e90b835df58ae046faa72f","source":{"kind":"arxiv","id":"1104.2303","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1104.2303","created_at":"2026-05-18T04:06:43Z"},{"alias_kind":"arxiv_version","alias_value":"1104.2303v3","created_at":"2026-05-18T04:06:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.2303","created_at":"2026-05-18T04:06:43Z"},{"alias_kind":"pith_short_12","alias_value":"4EAWMCN6UKPX","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_16","alias_value":"4EAWMCN6UKPXWKKY","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_8","alias_value":"4EAWMCN6","created_at":"2026-05-18T12:26:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:4EAWMCN6UKPXWKKY4I7RBLBIJN","target":"record","payload":{"canonical_record":{"source":{"id":"1104.2303","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2011-04-12T18:49:23Z","cross_cats_sorted":["cs.DM","math.NT"],"title_canon_sha256":"2d9cacf599b6a69feaa009462af81fd5144df32ff1dcc6bbb848c6840166a074","abstract_canon_sha256":"0c689197272f4aae93047a8efc245363cd14f5265dd5ab90491163a867e6b798"},"schema_version":"1.0"},"canonical_sha256":"e1016609bea29f7b2958e23f10ac284b59791de815e90b835df58ae046faa72f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:06:43.543746Z","signature_b64":"u5DAP2ZmuIGs7PSe3NG9l31lxVON98OSwlLn6U5S2wtrj7tsfNHhSuXdyI/LrwIkFj05riqOC66LaJYshq/CBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e1016609bea29f7b2958e23f10ac284b59791de815e90b835df58ae046faa72f","last_reissued_at":"2026-05-18T04:06:43.543121Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:06:43.543121Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1104.2303","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":"2026-05-18T04:06:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"liN02O1DtTWCQtcxQefvsabisvzL0ySPGLfLNA6wYyUsjD/LkSKZwKUnCv7Sdva5J75yd5S6QvjApGX9OTBuDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T03:44:43.856400Z"},"content_sha256":"9a262db866abff4e5bb966ae8395e904b5cdda4d4ddce8aea15bc9d39eb5a612","schema_version":"1.0","event_id":"sha256:9a262db866abff4e5bb966ae8395e904b5cdda4d4ddce8aea15bc9d39eb5a612"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:4EAWMCN6UKPXWKKY4I7RBLBIJN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Critical Exponent is Computable for Automatic Sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.NT"],"primary_cat":"cs.FL","authors_text":"Jeffrey Shallit, Luke Schaeffer","submitted_at":"2011-04-12T18:49:23Z","abstract_excerpt":"The critical exponent of an infinite word is defined to be the supremum of the exponent of each of its factors. For k-automatic sequences, we show that this critical exponent is always either a rational number or infinite, and its value is computable. Our results also apply to variants of the critical exponent, such as the initial critical exponent of Berthe, Holton, and Zamboni and the Diophantine exponent of Adamczewski and Bugeaud. Our work generalizes or recovers previous results of Krieger and others, and is applicable to other situations; e.g., the computation of the optimal recurrence c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.2303","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":""},"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":"2026-05-18T04:06:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XqpTTXYsrtOFbtGX6PlH5KBxFYbjlxTruCe283O24KVqUvtGRhP+MjwnqDrmVqd9rxQsUOt5OQPB3rHXVEzxAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T03:44:43.857074Z"},"content_sha256":"582fd8382549e2d27bc1826789302794290a27c49cf6956814c967076364b8be","schema_version":"1.0","event_id":"sha256:582fd8382549e2d27bc1826789302794290a27c49cf6956814c967076364b8be"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4EAWMCN6UKPXWKKY4I7RBLBIJN/bundle.json","state_url":"https://pith.science/pith/4EAWMCN6UKPXWKKY4I7RBLBIJN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4EAWMCN6UKPXWKKY4I7RBLBIJN/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":"2026-06-05T03:44:43Z","links":{"resolver":"https://pith.science/pith/4EAWMCN6UKPXWKKY4I7RBLBIJN","bundle":"https://pith.science/pith/4EAWMCN6UKPXWKKY4I7RBLBIJN/bundle.json","state":"https://pith.science/pith/4EAWMCN6UKPXWKKY4I7RBLBIJN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4EAWMCN6UKPXWKKY4I7RBLBIJN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:4EAWMCN6UKPXWKKY4I7RBLBIJN","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":"0c689197272f4aae93047a8efc245363cd14f5265dd5ab90491163a867e6b798","cross_cats_sorted":["cs.DM","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2011-04-12T18:49:23Z","title_canon_sha256":"2d9cacf599b6a69feaa009462af81fd5144df32ff1dcc6bbb848c6840166a074"},"schema_version":"1.0","source":{"id":"1104.2303","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1104.2303","created_at":"2026-05-18T04:06:43Z"},{"alias_kind":"arxiv_version","alias_value":"1104.2303v3","created_at":"2026-05-18T04:06:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.2303","created_at":"2026-05-18T04:06:43Z"},{"alias_kind":"pith_short_12","alias_value":"4EAWMCN6UKPX","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_16","alias_value":"4EAWMCN6UKPXWKKY","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_8","alias_value":"4EAWMCN6","created_at":"2026-05-18T12:26:20Z"}],"graph_snapshots":[{"event_id":"sha256:582fd8382549e2d27bc1826789302794290a27c49cf6956814c967076364b8be","target":"graph","created_at":"2026-05-18T04:06:43Z","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 critical exponent of an infinite word is defined to be the supremum of the exponent of each of its factors. For k-automatic sequences, we show that this critical exponent is always either a rational number or infinite, and its value is computable. Our results also apply to variants of the critical exponent, such as the initial critical exponent of Berthe, Holton, and Zamboni and the Diophantine exponent of Adamczewski and Bugeaud. Our work generalizes or recovers previous results of Krieger and others, and is applicable to other situations; e.g., the computation of the optimal recurrence c","authors_text":"Jeffrey Shallit, Luke Schaeffer","cross_cats":["cs.DM","math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2011-04-12T18:49:23Z","title":"The Critical Exponent is Computable for Automatic Sequences"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.2303","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":"sha256:9a262db866abff4e5bb966ae8395e904b5cdda4d4ddce8aea15bc9d39eb5a612","target":"record","created_at":"2026-05-18T04:06:43Z","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":"0c689197272f4aae93047a8efc245363cd14f5265dd5ab90491163a867e6b798","cross_cats_sorted":["cs.DM","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2011-04-12T18:49:23Z","title_canon_sha256":"2d9cacf599b6a69feaa009462af81fd5144df32ff1dcc6bbb848c6840166a074"},"schema_version":"1.0","source":{"id":"1104.2303","kind":"arxiv","version":3}},"canonical_sha256":"e1016609bea29f7b2958e23f10ac284b59791de815e90b835df58ae046faa72f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e1016609bea29f7b2958e23f10ac284b59791de815e90b835df58ae046faa72f","first_computed_at":"2026-05-18T04:06:43.543121Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:06:43.543121Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"u5DAP2ZmuIGs7PSe3NG9l31lxVON98OSwlLn6U5S2wtrj7tsfNHhSuXdyI/LrwIkFj05riqOC66LaJYshq/CBw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:06:43.543746Z","signed_message":"canonical_sha256_bytes"},"source_id":"1104.2303","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9a262db866abff4e5bb966ae8395e904b5cdda4d4ddce8aea15bc9d39eb5a612","sha256:582fd8382549e2d27bc1826789302794290a27c49cf6956814c967076364b8be"],"state_sha256":"6297df718f05cc29742fc8e4c76e753736fe2ba9d8f80518ae234f2c9e50c4cb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"658ZswxulQSzwE00uDoeMa9/iYqfJ22zGcIAAdKlYjvdbt6vm1uuQfS/6DHHD33zxofSFwqQF87oDogmNUOpBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T03:44:43.861116Z","bundle_sha256":"4f24380022dfd3b73d8fc835689b8191436b582e60e0abaf3794347dd42dd4ec"}}