{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:HE7O7D53M4MRZ3DZXUAGXAH4RA","short_pith_number":"pith:HE7O7D53","canonical_record":{"source":{"id":"1111.3380","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-11-14T22:03:10Z","cross_cats_sorted":[],"title_canon_sha256":"0b95d853953b20a318cf679b5767ed6bd84e8a5a3b16788ed8c2c68c4d8e976c","abstract_canon_sha256":"bcbf6b6d79899aa943c9be435d284c8f1fe0cc46044ecaba878dd3a998ea1c1f"},"schema_version":"1.0"},"canonical_sha256":"393eef8fbb67191cec79bd006b80fc881e295f0230729492ee1c33106784bc05","source":{"kind":"arxiv","id":"1111.3380","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.3380","created_at":"2026-05-18T03:52:26Z"},{"alias_kind":"arxiv_version","alias_value":"1111.3380v2","created_at":"2026-05-18T03:52:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.3380","created_at":"2026-05-18T03:52:26Z"},{"alias_kind":"pith_short_12","alias_value":"HE7O7D53M4MR","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_16","alias_value":"HE7O7D53M4MRZ3DZ","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_8","alias_value":"HE7O7D53","created_at":"2026-05-18T12:26:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:HE7O7D53M4MRZ3DZXUAGXAH4RA","target":"record","payload":{"canonical_record":{"source":{"id":"1111.3380","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-11-14T22:03:10Z","cross_cats_sorted":[],"title_canon_sha256":"0b95d853953b20a318cf679b5767ed6bd84e8a5a3b16788ed8c2c68c4d8e976c","abstract_canon_sha256":"bcbf6b6d79899aa943c9be435d284c8f1fe0cc46044ecaba878dd3a998ea1c1f"},"schema_version":"1.0"},"canonical_sha256":"393eef8fbb67191cec79bd006b80fc881e295f0230729492ee1c33106784bc05","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:52:26.212888Z","signature_b64":"jHsV5Ar//HK/gPh3kuSCm7YU7nlZlZuiGJkVJ63Jxtsf7uNJnSFbRlFFZzqV8CcbL1sfF5uxDFUbYRXpaiYSAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"393eef8fbb67191cec79bd006b80fc881e295f0230729492ee1c33106784bc05","last_reissued_at":"2026-05-18T03:52:26.212263Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:52:26.212263Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1111.3380","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:52:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ForyHaTNpFj4eHRZ4l5ZN85ra7WfgiTr0XWTqHBrAuqKGV1W0qItKL3lWzZDVWcH19htrFq+UZaWbO1qzVhlBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T07:48:47.489646Z"},"content_sha256":"b9ac2a0eb70241822941b3e8559f490ac52e051f4a0eadc79efeda8c0b19ce30","schema_version":"1.0","event_id":"sha256:b9ac2a0eb70241822941b3e8559f490ac52e051f4a0eadc79efeda8c0b19ce30"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:HE7O7D53M4MRZ3DZXUAGXAH4RA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the differences between consecutive prime numbers, I","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"A. H. Ledoan, D. A. Goldston","submitted_at":"2011-11-14T22:03:10Z","abstract_excerpt":"We show by an inclusion-exclusion argument that the prime $k$-tuple conjecture of Hardy and Littlewood provides an asymptotic formula for the number of consecutive prime numbers which are a specified distance apart. This refines one aspect of a theorem of Gallagher that the prime $k$-tuple conjecture implies that the prime numbers are distributed in a Poisson distribution around their average spacing."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.3380","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"2026-05-18T03:52:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"InhLAWXF7Keutc5zrgiJiZPTpCBPkLxlZqYi3niz+iuGAR2tqWuds8TTIw8RDNsrQDhRPSmeQoYLi59C+xWWBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T07:48:47.489995Z"},"content_sha256":"89b28c581a98549413275a35a1597eaca1811b7422d5f3ac5efe7553fc7fbbda","schema_version":"1.0","event_id":"sha256:89b28c581a98549413275a35a1597eaca1811b7422d5f3ac5efe7553fc7fbbda"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HE7O7D53M4MRZ3DZXUAGXAH4RA/bundle.json","state_url":"https://pith.science/pith/HE7O7D53M4MRZ3DZXUAGXAH4RA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HE7O7D53M4MRZ3DZXUAGXAH4RA/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":"2026-06-05T07:48:47Z","links":{"resolver":"https://pith.science/pith/HE7O7D53M4MRZ3DZXUAGXAH4RA","bundle":"https://pith.science/pith/HE7O7D53M4MRZ3DZXUAGXAH4RA/bundle.json","state":"https://pith.science/pith/HE7O7D53M4MRZ3DZXUAGXAH4RA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HE7O7D53M4MRZ3DZXUAGXAH4RA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:HE7O7D53M4MRZ3DZXUAGXAH4RA","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":"bcbf6b6d79899aa943c9be435d284c8f1fe0cc46044ecaba878dd3a998ea1c1f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-11-14T22:03:10Z","title_canon_sha256":"0b95d853953b20a318cf679b5767ed6bd84e8a5a3b16788ed8c2c68c4d8e976c"},"schema_version":"1.0","source":{"id":"1111.3380","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.3380","created_at":"2026-05-18T03:52:26Z"},{"alias_kind":"arxiv_version","alias_value":"1111.3380v2","created_at":"2026-05-18T03:52:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.3380","created_at":"2026-05-18T03:52:26Z"},{"alias_kind":"pith_short_12","alias_value":"HE7O7D53M4MR","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_16","alias_value":"HE7O7D53M4MRZ3DZ","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_8","alias_value":"HE7O7D53","created_at":"2026-05-18T12:26:30Z"}],"graph_snapshots":[{"event_id":"sha256:89b28c581a98549413275a35a1597eaca1811b7422d5f3ac5efe7553fc7fbbda","target":"graph","created_at":"2026-05-18T03:52:26Z","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":"We show by an inclusion-exclusion argument that the prime $k$-tuple conjecture of Hardy and Littlewood provides an asymptotic formula for the number of consecutive prime numbers which are a specified distance apart. This refines one aspect of a theorem of Gallagher that the prime $k$-tuple conjecture implies that the prime numbers are distributed in a Poisson distribution around their average spacing.","authors_text":"A. H. Ledoan, D. A. Goldston","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-11-14T22:03:10Z","title":"On the differences between consecutive prime numbers, I"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.3380","kind":"arxiv","version":2},"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:b9ac2a0eb70241822941b3e8559f490ac52e051f4a0eadc79efeda8c0b19ce30","target":"record","created_at":"2026-05-18T03:52:26Z","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":"bcbf6b6d79899aa943c9be435d284c8f1fe0cc46044ecaba878dd3a998ea1c1f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-11-14T22:03:10Z","title_canon_sha256":"0b95d853953b20a318cf679b5767ed6bd84e8a5a3b16788ed8c2c68c4d8e976c"},"schema_version":"1.0","source":{"id":"1111.3380","kind":"arxiv","version":2}},"canonical_sha256":"393eef8fbb67191cec79bd006b80fc881e295f0230729492ee1c33106784bc05","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"393eef8fbb67191cec79bd006b80fc881e295f0230729492ee1c33106784bc05","first_computed_at":"2026-05-18T03:52:26.212263Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:52:26.212263Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jHsV5Ar//HK/gPh3kuSCm7YU7nlZlZuiGJkVJ63Jxtsf7uNJnSFbRlFFZzqV8CcbL1sfF5uxDFUbYRXpaiYSAg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:52:26.212888Z","signed_message":"canonical_sha256_bytes"},"source_id":"1111.3380","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b9ac2a0eb70241822941b3e8559f490ac52e051f4a0eadc79efeda8c0b19ce30","sha256:89b28c581a98549413275a35a1597eaca1811b7422d5f3ac5efe7553fc7fbbda"],"state_sha256":"4132b2f7dd3093af3c5ca52f7a363e7894b3eb15c6c90ec3558f436a7def9113"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6SvUJREagigmmmb3hmTtJT3f4HPczACODnglRrdxbqpBBuIoTxLeMRPQ7uU7Bow9pnQ8dNbW+tWz2BiqoQC3Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T07:48:47.493276Z","bundle_sha256":"928b6549b8706c3a21ea67b90c9e17c111b941e5f54139c9ccc08c69d6d9be3b"}}