{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:MF4QCUEAOK53WSCCS5RK6E4B6V","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":"1d3fdd16ea314f3159ca8eb7c866a99f63b8c2c559a8db1515de6c667d015b5d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-10-26T04:27:14Z","title_canon_sha256":"203ebbd03f1103f2066bd233dac616ebeb4cb4b123dac10f7f240b7b005de999"},"schema_version":"1.0","source":{"id":"1510.07362","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.07362","created_at":"2026-05-18T01:29:14Z"},{"alias_kind":"arxiv_version","alias_value":"1510.07362v1","created_at":"2026-05-18T01:29:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.07362","created_at":"2026-05-18T01:29:14Z"},{"alias_kind":"pith_short_12","alias_value":"MF4QCUEAOK53","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_16","alias_value":"MF4QCUEAOK53WSCC","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_8","alias_value":"MF4QCUEA","created_at":"2026-05-18T12:29:32Z"}],"graph_snapshots":[{"event_id":"sha256:b8a3df64f4ac3e65b54d2f45388a7a85eafdde5950ab805504e117536ad60490","target":"graph","created_at":"2026-05-18T01:29:14Z","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":"Let $a$ be a positive integer, and let $\\sigma(a)$ denote the least natural number $s$ such that an integer square lies between $s^2 a$ and $s^2 (a+1)$; let $\\tau_s(a)$ denote the number of such integer squares. The function $\\sigma(a)$ and the sequence $(\\tau_s(a))_{s \\in \\mathbb{Z}^+}$ are studied, and are observed to exhibit surprisingly chaotic behavior. Upper- and lower-bounds for $\\sigma(a)$ are derived, as are criteria for when they are sharp.","authors_text":"Michael Weiss","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-10-26T04:27:14Z","title":"On the Distribution of Rational Squares"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.07362","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:b4026db50c408a84701a0b6f2cbc2d096f335f61d59ef1e3a8a1d449a6deec1a","target":"record","created_at":"2026-05-18T01:29:14Z","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":"1d3fdd16ea314f3159ca8eb7c866a99f63b8c2c559a8db1515de6c667d015b5d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-10-26T04:27:14Z","title_canon_sha256":"203ebbd03f1103f2066bd233dac616ebeb4cb4b123dac10f7f240b7b005de999"},"schema_version":"1.0","source":{"id":"1510.07362","kind":"arxiv","version":1}},"canonical_sha256":"617901508072bbbb48429762af1381f54594c293c64359962d7cdaf4cd5a6971","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"617901508072bbbb48429762af1381f54594c293c64359962d7cdaf4cd5a6971","first_computed_at":"2026-05-18T01:29:14.279674Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:29:14.279674Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TInrTCl5r1U1RUfi6n/+rNQ1Q2tFXagDsOA0ylO5XDmc6q4WbSTm44+Vf+fq/nmd0xg7uBR1a20c53zYyFW1DQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:29:14.280223Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.07362","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b4026db50c408a84701a0b6f2cbc2d096f335f61d59ef1e3a8a1d449a6deec1a","sha256:b8a3df64f4ac3e65b54d2f45388a7a85eafdde5950ab805504e117536ad60490"],"state_sha256":"590ce5106140e8c8c31fe9fdfc286c5c141ab9830aa2a8c763cba61b7b33e624"}