{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:ZUT6BSSPHCX3UTTFFQPGHNZE2K","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":"05cc4a3b8ad157dd25a8624ad06d67ebc9f1fec224bd22e89a4c2ee273a28c6b","cross_cats_sorted":["cs.LO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2012-05-08T19:05:08Z","title_canon_sha256":"08cdf82c5076830fce189c5be7db7332ebf110191b3cce5d117a150056e0187d"},"schema_version":"1.0","source":{"id":"1205.1775","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1205.1775","created_at":"2026-05-18T03:28:42Z"},{"alias_kind":"arxiv_version","alias_value":"1205.1775v1","created_at":"2026-05-18T03:28:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.1775","created_at":"2026-05-18T03:28:42Z"},{"alias_kind":"pith_short_12","alias_value":"ZUT6BSSPHCX3","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_16","alias_value":"ZUT6BSSPHCX3UTTF","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_8","alias_value":"ZUT6BSSP","created_at":"2026-05-18T12:27:30Z"}],"graph_snapshots":[{"event_id":"sha256:b874e0c9c4d765aeb1c43401e5d1026ee5a4c6b07443726aa03e9e1d31c39e32","target":"graph","created_at":"2026-05-18T03:28:42Z","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 prove that the injectively omega-tree-automatic ordinals are the ordinals smaller than $\\omega^{\\omega^\\omega}$. Then we show that the injectively $\\omega^n$-automatic ordinals, where $n>0$ is an integer, are the ordinals smaller than $\\omega^{\\omega^n}$. This strengthens a recent result of Schlicht and Stephan who considered in [Schlicht-Stephan11] the subclasses of finite word $\\omega^n$-automatic ordinals. As a by-product we obtain that the hierarchy of injectively $\\omega^n$-automatic structures, n>0, which was considered in [Finkel-Todorcevic12], is strict.","authors_text":"IMJ), Olivier Finkel (ELM, Stevo Todorcevic (ELM","cross_cats":["cs.LO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2012-05-08T19:05:08Z","title":"Automatic Ordinals"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.1775","kind":"arxiv","version":1},"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":"sha256:41c4ecf9f3f29931e802e00fb8a0b00ad2649fab5a96a1d038ed0c3e29bafb83","target":"record","created_at":"2026-05-18T03:28:42Z","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":"05cc4a3b8ad157dd25a8624ad06d67ebc9f1fec224bd22e89a4c2ee273a28c6b","cross_cats_sorted":["cs.LO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2012-05-08T19:05:08Z","title_canon_sha256":"08cdf82c5076830fce189c5be7db7332ebf110191b3cce5d117a150056e0187d"},"schema_version":"1.0","source":{"id":"1205.1775","kind":"arxiv","version":1}},"canonical_sha256":"cd27e0ca4f38afba4e652c1e63b724d28630b54bc07497873f2e3688232fdedd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cd27e0ca4f38afba4e652c1e63b724d28630b54bc07497873f2e3688232fdedd","first_computed_at":"2026-05-18T03:28:42.883523Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:28:42.883523Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kQ7tJRdft7B+H9DcSakG7Xg9+RJ3c9SBqaDIEjyykd8aGvb2e3n9BQWFnedcuo0B3gK8ob8k7Z5XZyuQTwZSAw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:28:42.884138Z","signed_message":"canonical_sha256_bytes"},"source_id":"1205.1775","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:41c4ecf9f3f29931e802e00fb8a0b00ad2649fab5a96a1d038ed0c3e29bafb83","sha256:b874e0c9c4d765aeb1c43401e5d1026ee5a4c6b07443726aa03e9e1d31c39e32"],"state_sha256":"dbeed7d181bbcd6994bd98742f54b5cda670945089e57c92438d18006698ea21"}