{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:F5WXFP4W3AFCRUDLMADML573UX","short_pith_number":"pith:F5WXFP4W","canonical_record":{"source":{"id":"2605.30959","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-29T07:55:39Z","cross_cats_sorted":[],"title_canon_sha256":"aa75dfa5dbd5d9fb4f26cafef492d154f7de9b88f461b01f3199fceadd15f31d","abstract_canon_sha256":"4c0ddda476dbdaa55190e6ddf3b7789258e26aa214a4446bff0731625d3032c5"},"schema_version":"1.0"},"canonical_sha256":"2f6d72bf96d80a28d06b6006c5f7fba5f1a898831fbc1eb4c14b187ebed1ffab","source":{"kind":"arxiv","id":"2605.30959","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.30959","created_at":"2026-06-01T01:03:27Z"},{"alias_kind":"arxiv_version","alias_value":"2605.30959v1","created_at":"2026-06-01T01:03:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.30959","created_at":"2026-06-01T01:03:27Z"},{"alias_kind":"pith_short_12","alias_value":"F5WXFP4W3AFC","created_at":"2026-06-01T01:03:27Z"},{"alias_kind":"pith_short_16","alias_value":"F5WXFP4W3AFCRUDL","created_at":"2026-06-01T01:03:27Z"},{"alias_kind":"pith_short_8","alias_value":"F5WXFP4W","created_at":"2026-06-01T01:03:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:F5WXFP4W3AFCRUDLMADML573UX","target":"record","payload":{"canonical_record":{"source":{"id":"2605.30959","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-29T07:55:39Z","cross_cats_sorted":[],"title_canon_sha256":"aa75dfa5dbd5d9fb4f26cafef492d154f7de9b88f461b01f3199fceadd15f31d","abstract_canon_sha256":"4c0ddda476dbdaa55190e6ddf3b7789258e26aa214a4446bff0731625d3032c5"},"schema_version":"1.0"},"canonical_sha256":"2f6d72bf96d80a28d06b6006c5f7fba5f1a898831fbc1eb4c14b187ebed1ffab","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-01T01:03:27.499215Z","signature_b64":"/PW1Ab0mLXE1jlBeelKEEdKVXb0IzcsqBytbqFU8vvgkQxqk+6jlSbXE15eb1Vx5iLUEguczGwKRMSXsDScJBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2f6d72bf96d80a28d06b6006c5f7fba5f1a898831fbc1eb4c14b187ebed1ffab","last_reissued_at":"2026-06-01T01:03:27.498740Z","signature_status":"signed_v1","first_computed_at":"2026-06-01T01:03:27.498740Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.30959","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":"2026-06-01T01:03:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gfICdIcBrwGHFKV9aOEcMizMuMZn/G3O0ggzgNoPHWJU69CA4uLntOrB8VaspPa6Zxge7vohVjob/YXSAGQwCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T12:13:33.288635Z"},"content_sha256":"ddeb2767467b489cccc0d848bfd921616027650b50d954a7f2c45cf4b81fbc24","schema_version":"1.0","event_id":"sha256:ddeb2767467b489cccc0d848bfd921616027650b50d954a7f2c45cf4b81fbc24"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:F5WXFP4W3AFCRUDLMADML573UX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An Improved Lower Bound for the de Bruijn--Erd\\H{o}s Consecutive Gap Problem","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Samuel Korsky","submitted_at":"2026-05-29T07:55:39Z","abstract_excerpt":"Let $(x_n)_{n\\geq 1}$ be a sequence of distinct points on the unit circle. After the first $n$ points are inserted, the circle is divided into $n$ intervals. For a fixed integer $r\\geq 1$, let $M_n^{(r)}$ and $m_n^{(r)}$ denote respectively the largest and smallest total lengths of $r$ consecutive intervals. A theorem of de Bruijn and Erd\\H{o}s gives \\[\n  \\limsup_{n\\to\\infty}\\frac{M_n^{(r)}}{m_n^{(r)}}\\geq 1+\\frac1r . \\] The case $r=1$ is sharp and gives the classical factor $2$. The cases $r\\geq 2$ remain much less understood. We prove the improved lower bound \\[\n  \\limsup_{n\\to\\infty}\\frac{M"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.30959","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/2605.30959/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":"2026-06-01T01:03:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kuEs7wz+0O/ypAP/sjPcq6I/SBoZl/j1Wc8pils6TFC3JC06iFyDJYM7nG4mnQr9wUNOu51eJ+k6yAreQwWCCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T12:13:33.288994Z"},"content_sha256":"7423db674195df030431bdc92b34f3d68e73896c8d644f32360924a8cc20e087","schema_version":"1.0","event_id":"sha256:7423db674195df030431bdc92b34f3d68e73896c8d644f32360924a8cc20e087"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/F5WXFP4W3AFCRUDLMADML573UX/bundle.json","state_url":"https://pith.science/pith/F5WXFP4W3AFCRUDLMADML573UX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/F5WXFP4W3AFCRUDLMADML573UX/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":"2026-07-07T12:13:33Z","links":{"resolver":"https://pith.science/pith/F5WXFP4W3AFCRUDLMADML573UX","bundle":"https://pith.science/pith/F5WXFP4W3AFCRUDLMADML573UX/bundle.json","state":"https://pith.science/pith/F5WXFP4W3AFCRUDLMADML573UX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/F5WXFP4W3AFCRUDLMADML573UX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:F5WXFP4W3AFCRUDLMADML573UX","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":"4c0ddda476dbdaa55190e6ddf3b7789258e26aa214a4446bff0731625d3032c5","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-29T07:55:39Z","title_canon_sha256":"aa75dfa5dbd5d9fb4f26cafef492d154f7de9b88f461b01f3199fceadd15f31d"},"schema_version":"1.0","source":{"id":"2605.30959","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.30959","created_at":"2026-06-01T01:03:27Z"},{"alias_kind":"arxiv_version","alias_value":"2605.30959v1","created_at":"2026-06-01T01:03:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.30959","created_at":"2026-06-01T01:03:27Z"},{"alias_kind":"pith_short_12","alias_value":"F5WXFP4W3AFC","created_at":"2026-06-01T01:03:27Z"},{"alias_kind":"pith_short_16","alias_value":"F5WXFP4W3AFCRUDL","created_at":"2026-06-01T01:03:27Z"},{"alias_kind":"pith_short_8","alias_value":"F5WXFP4W","created_at":"2026-06-01T01:03:27Z"}],"graph_snapshots":[{"event_id":"sha256:7423db674195df030431bdc92b34f3d68e73896c8d644f32360924a8cc20e087","target":"graph","created_at":"2026-06-01T01:03:27Z","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/2605.30959/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $(x_n)_{n\\geq 1}$ be a sequence of distinct points on the unit circle. After the first $n$ points are inserted, the circle is divided into $n$ intervals. For a fixed integer $r\\geq 1$, let $M_n^{(r)}$ and $m_n^{(r)}$ denote respectively the largest and smallest total lengths of $r$ consecutive intervals. A theorem of de Bruijn and Erd\\H{o}s gives \\[\n  \\limsup_{n\\to\\infty}\\frac{M_n^{(r)}}{m_n^{(r)}}\\geq 1+\\frac1r . \\] The case $r=1$ is sharp and gives the classical factor $2$. The cases $r\\geq 2$ remain much less understood. We prove the improved lower bound \\[\n  \\limsup_{n\\to\\infty}\\frac{M","authors_text":"Samuel Korsky","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-29T07:55:39Z","title":"An Improved Lower Bound for the de Bruijn--Erd\\H{o}s Consecutive Gap Problem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.30959","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:ddeb2767467b489cccc0d848bfd921616027650b50d954a7f2c45cf4b81fbc24","target":"record","created_at":"2026-06-01T01:03:27Z","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":"4c0ddda476dbdaa55190e6ddf3b7789258e26aa214a4446bff0731625d3032c5","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-29T07:55:39Z","title_canon_sha256":"aa75dfa5dbd5d9fb4f26cafef492d154f7de9b88f461b01f3199fceadd15f31d"},"schema_version":"1.0","source":{"id":"2605.30959","kind":"arxiv","version":1}},"canonical_sha256":"2f6d72bf96d80a28d06b6006c5f7fba5f1a898831fbc1eb4c14b187ebed1ffab","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2f6d72bf96d80a28d06b6006c5f7fba5f1a898831fbc1eb4c14b187ebed1ffab","first_computed_at":"2026-06-01T01:03:27.498740Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-01T01:03:27.498740Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/PW1Ab0mLXE1jlBeelKEEdKVXb0IzcsqBytbqFU8vvgkQxqk+6jlSbXE15eb1Vx5iLUEguczGwKRMSXsDScJBg==","signature_status":"signed_v1","signed_at":"2026-06-01T01:03:27.499215Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.30959","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ddeb2767467b489cccc0d848bfd921616027650b50d954a7f2c45cf4b81fbc24","sha256:7423db674195df030431bdc92b34f3d68e73896c8d644f32360924a8cc20e087"],"state_sha256":"157e034d9b113b10986bca0b82973b0c61ad5cf33d23867ac0fd282e5924be5a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"q7gJ8YSJsD8/LkbyJ9TKwWFHCbDGI2LQyLUbsIS+0Q7SHHkkU+hP2Jne8dfyXCO8dNURlZI98WsjNvafm0pAAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-07T12:13:33.290881Z","bundle_sha256":"355004e4fb92e0217c6dbd00262b975c2023c35d95293defaa221f0a23f0030b"}}