{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:F73ZAJYOF6JZDSL3VXYHWSOCTU","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":"1f4180fc0c3ea617eab54da5c7cbf915153cbb18256f1907084186a0260b980c","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-01-27T00:01:03Z","title_canon_sha256":"b5b7bd075222a8dcd5d6d38a89128c5e2cd760b53a79574c438da405fc0ae824"},"schema_version":"1.0","source":{"id":"2501.15707","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2501.15707","created_at":"2026-07-05T10:05:33Z"},{"alias_kind":"arxiv_version","alias_value":"2501.15707v1","created_at":"2026-07-05T10:05:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2501.15707","created_at":"2026-07-05T10:05:33Z"},{"alias_kind":"pith_short_12","alias_value":"F73ZAJYOF6JZ","created_at":"2026-07-05T10:05:33Z"},{"alias_kind":"pith_short_16","alias_value":"F73ZAJYOF6JZDSL3","created_at":"2026-07-05T10:05:33Z"},{"alias_kind":"pith_short_8","alias_value":"F73ZAJYO","created_at":"2026-07-05T10:05:33Z"}],"graph_snapshots":[{"event_id":"sha256:3c1243b55d8e378ada4da42976c2eb2b733befffd4910f910c9658e0b23a55a4","target":"graph","created_at":"2026-07-05T10:05:33Z","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/2501.15707/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This paper investigates the existence of integers that exclude two specific residence values modulo primes up to $p_k$ within the interval $[p_k^2, p_{k+1}^2]$. Using asymptotic results from analytic number theory, we establish bounds on the proportion of integers excluded by the union of residue classes. The findings highlight the density of residue class coverage in large intervals, contributing to the understanding of modular systems and their implications in number theory and related fields.","authors_text":"Liang Zhao","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-01-27T00:01:03Z","title":"Proof of Existence of Integers Excluding Two Residue Values in a Specific Range"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.15707","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":"sha256:2491a1e961dcc317aeb2493fca79e7f94b36c85b9e031b15e94bd885eeef10cb","target":"record","created_at":"2026-07-05T10:05:33Z","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":"1f4180fc0c3ea617eab54da5c7cbf915153cbb18256f1907084186a0260b980c","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-01-27T00:01:03Z","title_canon_sha256":"b5b7bd075222a8dcd5d6d38a89128c5e2cd760b53a79574c438da405fc0ae824"},"schema_version":"1.0","source":{"id":"2501.15707","kind":"arxiv","version":1}},"canonical_sha256":"2ff790270e2f9391c97badf07b49c29d1271977efb3112ca0de2ada47138aa5a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2ff790270e2f9391c97badf07b49c29d1271977efb3112ca0de2ada47138aa5a","first_computed_at":"2026-07-05T10:05:33.171337Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:05:33.171337Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GlWikehXVUY890zYY0n5NSsVNFeBQ5kpj66NCwrx3uSOe5WueqbhS45bj3QwJTUdPWRXZ2TSlJ3whYbAa0gxCA==","signature_status":"signed_v1","signed_at":"2026-07-05T10:05:33.171854Z","signed_message":"canonical_sha256_bytes"},"source_id":"2501.15707","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2491a1e961dcc317aeb2493fca79e7f94b36c85b9e031b15e94bd885eeef10cb","sha256:3c1243b55d8e378ada4da42976c2eb2b733befffd4910f910c9658e0b23a55a4"],"state_sha256":"9f64d9015ee2f8f8e6665283b18bb5955aee85c4204bbf4634ac55d92aadc19b"}