{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:226XPPFONIRYPU2OVK57EWDAZT","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":"084df6085aadcb18cfd0e14ae297d40080a79360108966a2a0f5ac99d951776a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-10-27T20:00:35Z","title_canon_sha256":"2c3a556c181674aba17c6ac8b7c358206056919007d944a05e108e93f3def3ca"},"schema_version":"1.0","source":{"id":"1510.08054","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.08054","created_at":"2026-05-18T01:28:44Z"},{"alias_kind":"arxiv_version","alias_value":"1510.08054v1","created_at":"2026-05-18T01:28:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.08054","created_at":"2026-05-18T01:28:44Z"},{"alias_kind":"pith_short_12","alias_value":"226XPPFONIRY","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_16","alias_value":"226XPPFONIRYPU2O","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_8","alias_value":"226XPPFO","created_at":"2026-05-18T12:28:59Z"}],"graph_snapshots":[{"event_id":"sha256:b62501fab2ce952cad23b50100e042539cfd69f416a725771d2c57bc094144a2","target":"graph","created_at":"2026-05-18T01:28:44Z","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 $d_n = p_{n+1} - p_n$, where $p_n$ denotes the $n$th smallest prime, and let $R(T) = \\log T \\log_2 T\\log_4 T/(\\log_3 T)^2$ (the \"Erd{\\H o}s--Rankin\" function). We consider the sequence $(d_n/R(p_n))$ of normalized prime gaps, and show that its limit point set contains at least $25\\%$ of nonnegative real numbers. We also show that the same result holds if $R(T)$ is replaced by any \"reasonable\" function that tends to infinity more slowly than $R(T)\\log_3 T$. We also consider \"chains\" of normalized prime gaps. Our proof combines breakthrough work of Maynard and Tao on bounded gaps between pri","authors_text":"Roger Baker, Tristan Freiberg","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-10-27T20:00:35Z","title":"Limit points and long gaps between primes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.08054","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:aeb210c4f00eedf9c18ef0004e05c6459f75e2d23c370437f0cce4aab4c9835b","target":"record","created_at":"2026-05-18T01:28:44Z","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":"084df6085aadcb18cfd0e14ae297d40080a79360108966a2a0f5ac99d951776a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-10-27T20:00:35Z","title_canon_sha256":"2c3a556c181674aba17c6ac8b7c358206056919007d944a05e108e93f3def3ca"},"schema_version":"1.0","source":{"id":"1510.08054","kind":"arxiv","version":1}},"canonical_sha256":"d6bd77bcae6a2387d34eaabbf25860ccfb42b850dc049d57739841d4787c974e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d6bd77bcae6a2387d34eaabbf25860ccfb42b850dc049d57739841d4787c974e","first_computed_at":"2026-05-18T01:28:44.645502Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:28:44.645502Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LKuFFwNTsZAX6UQQ/9ZM6g6P+vJ2wQqjxwxGG47VwLV05OdNHdjD5bgE5v+JeU/mmp6PG8mxykyER7pTjGjbBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:28:44.646038Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.08054","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:aeb210c4f00eedf9c18ef0004e05c6459f75e2d23c370437f0cce4aab4c9835b","sha256:b62501fab2ce952cad23b50100e042539cfd69f416a725771d2c57bc094144a2"],"state_sha256":"25dcc1f39cf24e0ca032789b991a68e4cea4776ba6630ae263b2500cdb5c8d66"}