{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:OFVNG5WLMVCA4QS72EGYSDWEHO","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":"30ace8a42ca9e81787d67b4004611eef3fdeb00461dd0685af2a94b80f3eb972","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2024-06-03T23:09:12Z","title_canon_sha256":"d57d7480b404c86aa015a2b9a414b234fffced6741bbf49fa60ba85ce93139c7"},"schema_version":"1.0","source":{"id":"2406.01836","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2406.01836","created_at":"2026-07-05T08:26:49Z"},{"alias_kind":"arxiv_version","alias_value":"2406.01836v1","created_at":"2026-07-05T08:26:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2406.01836","created_at":"2026-07-05T08:26:49Z"},{"alias_kind":"pith_short_12","alias_value":"OFVNG5WLMVCA","created_at":"2026-07-05T08:26:49Z"},{"alias_kind":"pith_short_16","alias_value":"OFVNG5WLMVCA4QS7","created_at":"2026-07-05T08:26:49Z"},{"alias_kind":"pith_short_8","alias_value":"OFVNG5WL","created_at":"2026-07-05T08:26:49Z"}],"graph_snapshots":[{"event_id":"sha256:18ac50060d379602228b4c3e2906d7b9e82ca57633c7f8fd559328ec1825058c","target":"graph","created_at":"2026-07-05T08:26:49Z","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/2406.01836/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For any limit ordinal $\\lambda$, we construct a linear order $L_\\lambda$ whose Scott complexity is $\\Sigma_{\\lambda+1}$. This completes the classification of the possible Scott sentence complexities of linear orderings. Previously, there was only one known construction of any structure (of any signature) with Scott complexity $\\Sigma_{\\lambda+1}$, and our construction gives new examples, e.g., rigid structures, of this complexity.\n  Moreover, we can construct the linear orders $L_\\lambda$ so that not only does $L_\\lambda$ have Scott complexity $\\Sigma_{\\lambda+1}$, but there are continuum-many","authors_text":"David Gonzalez, Matthew Harrison-Trainor, Meng-Che \"Turbo\" Ho","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2024-06-03T23:09:12Z","title":"Scott analysis, linear orders and almost periodic functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2406.01836","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:fec081f48344754d6b9d0cce817f4b37f0ba7d12c8a1caac78a6bef773b122fb","target":"record","created_at":"2026-07-05T08:26:49Z","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":"30ace8a42ca9e81787d67b4004611eef3fdeb00461dd0685af2a94b80f3eb972","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2024-06-03T23:09:12Z","title_canon_sha256":"d57d7480b404c86aa015a2b9a414b234fffced6741bbf49fa60ba85ce93139c7"},"schema_version":"1.0","source":{"id":"2406.01836","kind":"arxiv","version":1}},"canonical_sha256":"716ad376cb65440e425fd10d890ec43bb34d313b2ee60251d125b6d14a17a75f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"716ad376cb65440e425fd10d890ec43bb34d313b2ee60251d125b6d14a17a75f","first_computed_at":"2026-07-05T08:26:49.296740Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:26:49.296740Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ycIWCiBzLhG9o8uF5Y2Q7umCTYL/l/pVH36IGBGem6URHLz/cEtxxEpOjxrLJWyddhxPkhwYlcOMJb7WSGlzDw==","signature_status":"signed_v1","signed_at":"2026-07-05T08:26:49.297146Z","signed_message":"canonical_sha256_bytes"},"source_id":"2406.01836","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fec081f48344754d6b9d0cce817f4b37f0ba7d12c8a1caac78a6bef773b122fb","sha256:18ac50060d379602228b4c3e2906d7b9e82ca57633c7f8fd559328ec1825058c"],"state_sha256":"61b2f209a7b0f75ec8cbe387e73da4ccaca9da94fbc3e36c02c63accdf4ffaec"}