{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:VHPUZX446DDAKLUDTEPM7EEIUH","short_pith_number":"pith:VHPUZX44","canonical_record":{"source":{"id":"2607.06551","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-07T17:53:58Z","cross_cats_sorted":[],"title_canon_sha256":"22560ab113ea16fbe54859b68ea4c699c09185ab3ad9443d5f2c0846570950dc","abstract_canon_sha256":"0de88dbfb4facc0bda0010546cd03f3764ac1b40d81a201d4c7ac690bb9c8166"},"schema_version":"1.0"},"canonical_sha256":"a9df4cdf9cf0c6052e83991ecf9088a1eb2149175e8e7996baa2484ba7c14e57","source":{"kind":"arxiv","id":"2607.06551","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.06551","created_at":"2026-07-08T01:19:29Z"},{"alias_kind":"arxiv_version","alias_value":"2607.06551v1","created_at":"2026-07-08T01:19:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.06551","created_at":"2026-07-08T01:19:29Z"},{"alias_kind":"pith_short_12","alias_value":"VHPUZX446DDA","created_at":"2026-07-08T01:19:29Z"},{"alias_kind":"pith_short_16","alias_value":"VHPUZX446DDAKLUD","created_at":"2026-07-08T01:19:29Z"},{"alias_kind":"pith_short_8","alias_value":"VHPUZX44","created_at":"2026-07-08T01:19:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:VHPUZX446DDAKLUDTEPM7EEIUH","target":"record","payload":{"canonical_record":{"source":{"id":"2607.06551","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-07T17:53:58Z","cross_cats_sorted":[],"title_canon_sha256":"22560ab113ea16fbe54859b68ea4c699c09185ab3ad9443d5f2c0846570950dc","abstract_canon_sha256":"0de88dbfb4facc0bda0010546cd03f3764ac1b40d81a201d4c7ac690bb9c8166"},"schema_version":"1.0"},"canonical_sha256":"a9df4cdf9cf0c6052e83991ecf9088a1eb2149175e8e7996baa2484ba7c14e57","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-08T01:19:29.676092Z","signature_b64":"KLSacLLwba8rk7uJw3sjREdQna4lgXsM06Nyu1TzUgH9gPgcHEuYdv9qd1dF9OgTwbreGbIhKYI4NGk4TiEwDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a9df4cdf9cf0c6052e83991ecf9088a1eb2149175e8e7996baa2484ba7c14e57","last_reissued_at":"2026-07-08T01:19:29.675653Z","signature_status":"signed_v1","first_computed_at":"2026-07-08T01:19:29.675653Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.06551","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":"2026-07-08T01:19:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cASZZcweV5OOcxEoG0oetNVblbV7K5uLSczWmrg5Wu1h0GaLykHGY3oDc8OUmo3v3Cc+/zAL26LoOZ0smHCMBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-02T17:02:33.891766Z"},"content_sha256":"4d2e81804644d038fd0d79b836dad30013d87d437280f1f933ba24c6b70c74d6","schema_version":"1.0","event_id":"sha256:4d2e81804644d038fd0d79b836dad30013d87d437280f1f933ba24c6b70c74d6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:VHPUZX446DDAKLUDTEPM7EEIUH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Tight Staircase Bounds for Cyclic Subsets below Dirac's Threshold","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hong Liu, Lanchao Wang, Mengyuan Niu, Zhifei Yan","submitted_at":"2026-07-07T17:53:58Z","abstract_excerpt":"Let $\\operatorname{Cyc}(G)$ denote the number of cyclic subsets in a graph $G$, which are subsets that induce a Hamiltonian subgraph. Dragani\\'{c}, Keevash and M\\\"{u}yesser recently proved that every regular Dirac graph has $\\Omega(2^n)$ cyclic subsets, resolving a problem of Erd\\H{o}s and Faudree.\n  We determine the sharp asymptotic lower bound throughout the linear range below Dirac's threshold. Let $G$ be an $n$-vertex $d$-regular graph with $d=\\Omega(n)$ and $d<n/2$, then $$\n  \\operatorname{Cyc}(G)\\ge (q-o(1))2^{n/q}, \\quad \\text{where } \\quad q=\\left\\lfloor \\frac{n}{d+1}\\right\\rfloor \\ge "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.06551","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/2607.06551/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":"2026-07-08T01:19:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FqQPnKD/vDYRFbuizd3+bA2j5YPE10FheNbK0KDXEWSzuK3+0snj7TQOrdEQJQi7kAhYLyGZIfbhbgjIvzMZCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-02T17:02:33.892328Z"},"content_sha256":"e8973ed1b0b5b71c4fb8a2525e49463ab5175e37c765043a0a72b2433fce2940","schema_version":"1.0","event_id":"sha256:e8973ed1b0b5b71c4fb8a2525e49463ab5175e37c765043a0a72b2433fce2940"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VHPUZX446DDAKLUDTEPM7EEIUH/bundle.json","state_url":"https://pith.science/pith/VHPUZX446DDAKLUDTEPM7EEIUH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VHPUZX446DDAKLUDTEPM7EEIUH/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":"2026-08-02T17:02:33Z","links":{"resolver":"https://pith.science/pith/VHPUZX446DDAKLUDTEPM7EEIUH","bundle":"https://pith.science/pith/VHPUZX446DDAKLUDTEPM7EEIUH/bundle.json","state":"https://pith.science/pith/VHPUZX446DDAKLUDTEPM7EEIUH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VHPUZX446DDAKLUDTEPM7EEIUH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:VHPUZX446DDAKLUDTEPM7EEIUH","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":"0de88dbfb4facc0bda0010546cd03f3764ac1b40d81a201d4c7ac690bb9c8166","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-07T17:53:58Z","title_canon_sha256":"22560ab113ea16fbe54859b68ea4c699c09185ab3ad9443d5f2c0846570950dc"},"schema_version":"1.0","source":{"id":"2607.06551","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.06551","created_at":"2026-07-08T01:19:29Z"},{"alias_kind":"arxiv_version","alias_value":"2607.06551v1","created_at":"2026-07-08T01:19:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.06551","created_at":"2026-07-08T01:19:29Z"},{"alias_kind":"pith_short_12","alias_value":"VHPUZX446DDA","created_at":"2026-07-08T01:19:29Z"},{"alias_kind":"pith_short_16","alias_value":"VHPUZX446DDAKLUD","created_at":"2026-07-08T01:19:29Z"},{"alias_kind":"pith_short_8","alias_value":"VHPUZX44","created_at":"2026-07-08T01:19:29Z"}],"graph_snapshots":[{"event_id":"sha256:e8973ed1b0b5b71c4fb8a2525e49463ab5175e37c765043a0a72b2433fce2940","target":"graph","created_at":"2026-07-08T01:19:29Z","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/2607.06551/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $\\operatorname{Cyc}(G)$ denote the number of cyclic subsets in a graph $G$, which are subsets that induce a Hamiltonian subgraph. Dragani\\'{c}, Keevash and M\\\"{u}yesser recently proved that every regular Dirac graph has $\\Omega(2^n)$ cyclic subsets, resolving a problem of Erd\\H{o}s and Faudree.\n  We determine the sharp asymptotic lower bound throughout the linear range below Dirac's threshold. Let $G$ be an $n$-vertex $d$-regular graph with $d=\\Omega(n)$ and $d<n/2$, then $$\n  \\operatorname{Cyc}(G)\\ge (q-o(1))2^{n/q}, \\quad \\text{where } \\quad q=\\left\\lfloor \\frac{n}{d+1}\\right\\rfloor \\ge ","authors_text":"Hong Liu, Lanchao Wang, Mengyuan Niu, Zhifei Yan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-07T17:53:58Z","title":"Tight Staircase Bounds for Cyclic Subsets below Dirac's Threshold"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.06551","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":"sha256:4d2e81804644d038fd0d79b836dad30013d87d437280f1f933ba24c6b70c74d6","target":"record","created_at":"2026-07-08T01:19:29Z","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":"0de88dbfb4facc0bda0010546cd03f3764ac1b40d81a201d4c7ac690bb9c8166","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-07T17:53:58Z","title_canon_sha256":"22560ab113ea16fbe54859b68ea4c699c09185ab3ad9443d5f2c0846570950dc"},"schema_version":"1.0","source":{"id":"2607.06551","kind":"arxiv","version":1}},"canonical_sha256":"a9df4cdf9cf0c6052e83991ecf9088a1eb2149175e8e7996baa2484ba7c14e57","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a9df4cdf9cf0c6052e83991ecf9088a1eb2149175e8e7996baa2484ba7c14e57","first_computed_at":"2026-07-08T01:19:29.675653Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-08T01:19:29.675653Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KLSacLLwba8rk7uJw3sjREdQna4lgXsM06Nyu1TzUgH9gPgcHEuYdv9qd1dF9OgTwbreGbIhKYI4NGk4TiEwDA==","signature_status":"signed_v1","signed_at":"2026-07-08T01:19:29.676092Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.06551","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4d2e81804644d038fd0d79b836dad30013d87d437280f1f933ba24c6b70c74d6","sha256:e8973ed1b0b5b71c4fb8a2525e49463ab5175e37c765043a0a72b2433fce2940"],"state_sha256":"f84fbacca32935afaf27dfeda33df9be708a8d48148822c0dade2c91eabf6a6c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hVUsTapkbgDKG70DhpQhSTXibT1kBoaK1vS0JLekAxUNNc6JWwvzAwET0umGBl9EeCNG+H7zztLnYwzMTVG2BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-02T17:02:33.899665Z","bundle_sha256":"a2b63ccbe672f4f4d5c95480ad328675df92176005fcbd331e97a17ff141b781"}}