{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:PXEFDQBQ4PAMOGTI4NDKHVYLNZ","short_pith_number":"pith:PXEFDQBQ","canonical_record":{"source":{"id":"2607.12817","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-14T14:30:17Z","cross_cats_sorted":[],"title_canon_sha256":"b211caf36d44bd5c18a009e9c16372bcb447fa2979435003e013b5852029345b","abstract_canon_sha256":"003acfc8386bf9666d481ebf54ba21049c6fc8486b1b5f37fa1a3aa574be3886"},"schema_version":"1.0"},"canonical_sha256":"7dc851c030e3c0c71a68e346a3d70b6e635e69f150e3ee773c738317c7e55c18","source":{"kind":"arxiv","id":"2607.12817","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"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:PXEFDQBQ4PAMOGTI4NDKHVYLNZ","target":"record","payload":{"canonical_record":{"source":{"id":"2607.12817","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-14T14:30:17Z","cross_cats_sorted":[],"title_canon_sha256":"b211caf36d44bd5c18a009e9c16372bcb447fa2979435003e013b5852029345b","abstract_canon_sha256":"003acfc8386bf9666d481ebf54ba21049c6fc8486b1b5f37fa1a3aa574be3886"},"schema_version":"1.0"},"canonical_sha256":"7dc851c030e3c0c71a68e346a3d70b6e635e69f150e3ee773c738317c7e55c18","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-15T01:22:19.551685Z","signature_b64":"5efzK/IuFFSJS7evqaSFRWrdpr1z3CFKg/rNzQcDUJ5PSXxdEyjskNxizzNWs5Y7Sq7QN2YwgifYVtOswvJJDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7dc851c030e3c0c71a68e346a3d70b6e635e69f150e3ee773c738317c7e55c18","last_reissued_at":"2026-07-15T01:22:19.550726Z","signature_status":"signed_v1","first_computed_at":"2026-07-15T01:22:19.550726Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.12817","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-15T01:22:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pdOaC47Hm0fMg6T+qGGn/nHv+fEwxqDcGjSm0ZeA5wWCFD1ZJ3I01BE4TI3jmHHaswbNQr8w/3x9FfPd3lYUBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T22:17:49.087527Z"},"content_sha256":"b1dbc9caf749ac627fd176bb9cf67230e7f496bbe1ef68f7cd7d88cac36778f1","schema_version":"1.0","event_id":"sha256:b1dbc9caf749ac627fd176bb9cf67230e7f496bbe1ef68f7cd7d88cac36778f1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:PXEFDQBQ4PAMOGTI4NDKHVYLNZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Note on Iterated Beatty Sequences","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Afshin Zarei, Ali N. Valizadeh, Mohsen Khani","submitted_at":"2026-07-14T14:30:17Z","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 $\\alpha$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.12817","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.12817/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-15T01:22:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RwHZYi0lTWzpx+tD3pKqfuts337bCd0pggpFDe28mqpKCXECHcGM2KbnwbKWaeN+q3v+Ha3V3XCEUruvoLRWAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T22:17:49.088059Z"},"content_sha256":"010695abc9f61ed530e116c7a0536660d5491a64fc6b0869d1849375104bd0d8","schema_version":"1.0","event_id":"sha256:010695abc9f61ed530e116c7a0536660d5491a64fc6b0869d1849375104bd0d8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PXEFDQBQ4PAMOGTI4NDKHVYLNZ/bundle.json","state_url":"https://pith.science/pith/PXEFDQBQ4PAMOGTI4NDKHVYLNZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PXEFDQBQ4PAMOGTI4NDKHVYLNZ/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":"2026-08-10T22:17:49Z","links":{"resolver":"https://pith.science/pith/PXEFDQBQ4PAMOGTI4NDKHVYLNZ","bundle":"https://pith.science/pith/PXEFDQBQ4PAMOGTI4NDKHVYLNZ/bundle.json","state":"https://pith.science/pith/PXEFDQBQ4PAMOGTI4NDKHVYLNZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PXEFDQBQ4PAMOGTI4NDKHVYLNZ/bundle.json"},"state":{"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"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BJ5FLaJXQ2+yRNiRMNicyq9tE/ORPMcMM8U47tGpuAgQozhc0EOM7AdaZ+vxb3vT/wfKRSO7Hm6dzxCCv7MDBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-10T22:17:49.094347Z","bundle_sha256":"54025192f1f0e5fc44cbb44ba807b02c9f31773c8242e8780fd43b9949f978a7"}}