{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:PXEFDQBQ4PAMOGTI4NDKHVYLNZ","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":"003acfc8386bf9666d481ebf54ba21049c6fc8486b1b5f37fa1a3aa574be3886","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-14T14:30:17Z","title_canon_sha256":"b211caf36d44bd5c18a009e9c16372bcb447fa2979435003e013b5852029345b"},"schema_version":"1.0","source":{"id":"2607.12817","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.12817","created_at":"2026-07-15T01:22:19Z"},{"alias_kind":"arxiv_version","alias_value":"2607.12817v1","created_at":"2026-07-15T01:22:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.12817","created_at":"2026-07-15T01:22:19Z"},{"alias_kind":"pith_short_12","alias_value":"PXEFDQBQ4PAM","created_at":"2026-07-15T01:22:19Z"},{"alias_kind":"pith_short_16","alias_value":"PXEFDQBQ4PAMOGTI","created_at":"2026-07-15T01:22:19Z"},{"alias_kind":"pith_short_8","alias_value":"PXEFDQBQ","created_at":"2026-07-15T01:22:19Z"}],"graph_snapshots":[{"event_id":"sha256:010695abc9f61ed530e116c7a0536660d5491a64fc6b0869d1849375104bd0d8","target":"graph","created_at":"2026-07-15T01:22:19Z","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/2607.12817/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For any irrational number $ \\alpha>\\frac{3+\\sqrt{5}}{2}\\approx2.618$ and given a positive $ n\\in\\mathbb{N} $, we use elementary number theory to introduce a necessary and sufficient condition for a natural number $ x $ to be in the $n$th iterate of the Beatty sequence of modulus $\\alpha$.","authors_text":"Afshin Zarei, Ali N. Valizadeh, Mohsen Khani","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-14T14:30:17Z","title":"A Note on Iterated Beatty Sequences"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.12817","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:b1dbc9caf749ac627fd176bb9cf67230e7f496bbe1ef68f7cd7d88cac36778f1","target":"record","created_at":"2026-07-15T01:22:19Z","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":"003acfc8386bf9666d481ebf54ba21049c6fc8486b1b5f37fa1a3aa574be3886","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-14T14:30:17Z","title_canon_sha256":"b211caf36d44bd5c18a009e9c16372bcb447fa2979435003e013b5852029345b"},"schema_version":"1.0","source":{"id":"2607.12817","kind":"arxiv","version":1}},"canonical_sha256":"7dc851c030e3c0c71a68e346a3d70b6e635e69f150e3ee773c738317c7e55c18","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7dc851c030e3c0c71a68e346a3d70b6e635e69f150e3ee773c738317c7e55c18","first_computed_at":"2026-07-15T01:22:19.550726Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-15T01:22:19.550726Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5efzK/IuFFSJS7evqaSFRWrdpr1z3CFKg/rNzQcDUJ5PSXxdEyjskNxizzNWs5Y7Sq7QN2YwgifYVtOswvJJDg==","signature_status":"signed_v1","signed_at":"2026-07-15T01:22:19.551685Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.12817","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b1dbc9caf749ac627fd176bb9cf67230e7f496bbe1ef68f7cd7d88cac36778f1","sha256:010695abc9f61ed530e116c7a0536660d5491a64fc6b0869d1849375104bd0d8"],"state_sha256":"4b28b0c80320fdeb54938f0e6bf47f9380db43c46b3f141e28495541cccad2f4"}