{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:TOODYODFSJD5DVFEHYLXG4MTUA","short_pith_number":"pith:TOODYODF","canonical_record":{"source":{"id":"1608.04834","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-08-17T02:12:14Z","cross_cats_sorted":["cs.NA","math.CA","math.CO"],"title_canon_sha256":"948c3bc467ce950dd8eb073ae1194d806dda4c359b4da7aa011a7c9483f99906","abstract_canon_sha256":"33b019b582a8594dfaec3104bf45313ac32dbead8ba1f3e7e15e68205b5691fe"},"schema_version":"1.0"},"canonical_sha256":"9b9c3c38659247d1d4a43e17737193a03023bef7ab4d1e22171380fa68dcd87a","source":{"kind":"arxiv","id":"1608.04834","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.04834","created_at":"2026-07-05T03:41:55Z"},{"alias_kind":"arxiv_version","alias_value":"1608.04834v4","created_at":"2026-07-05T03:41:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.04834","created_at":"2026-07-05T03:41:55Z"},{"alias_kind":"pith_short_12","alias_value":"TOODYODFSJD5","created_at":"2026-07-05T03:41:55Z"},{"alias_kind":"pith_short_16","alias_value":"TOODYODFSJD5DVFE","created_at":"2026-07-05T03:41:55Z"},{"alias_kind":"pith_short_8","alias_value":"TOODYODF","created_at":"2026-07-05T03:41:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:TOODYODFSJD5DVFEHYLXG4MTUA","target":"record","payload":{"canonical_record":{"source":{"id":"1608.04834","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-08-17T02:12:14Z","cross_cats_sorted":["cs.NA","math.CA","math.CO"],"title_canon_sha256":"948c3bc467ce950dd8eb073ae1194d806dda4c359b4da7aa011a7c9483f99906","abstract_canon_sha256":"33b019b582a8594dfaec3104bf45313ac32dbead8ba1f3e7e15e68205b5691fe"},"schema_version":"1.0"},"canonical_sha256":"9b9c3c38659247d1d4a43e17737193a03023bef7ab4d1e22171380fa68dcd87a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:41:55.472889Z","signature_b64":"/Znpjm06+QB9/lC+fFV4lT5GSUjmLDi0zrpyaU0wKwkXqj56sp8h3gCyn92y9XrrJyjZZOONBpbKDhXFZABGCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9b9c3c38659247d1d4a43e17737193a03023bef7ab4d1e22171380fa68dcd87a","last_reissued_at":"2026-07-05T03:41:55.472538Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:41:55.472538Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1608.04834","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T03:41:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YDbJw12mWoFTCDyq4hJzRAgYZy94HylizWQicVPqI8i24gVI5JFZbN7+cTPozFZ8Gy7eqQS8Vma3+GX2c5JhAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T18:47:03.523857Z"},"content_sha256":"64c15f77d97e7afca7a9e9645542620f221a6d99023651c2b682af91742b4417","schema_version":"1.0","event_id":"sha256:64c15f77d97e7afca7a9e9645542620f221a6d99023651c2b682af91742b4417"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:TOODYODFSJD5DVFEHYLXG4MTUA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Asymptotic approximation of central binomial coefficients with rigorous error bounds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA","math.CA","math.CO"],"primary_cat":"math.NA","authors_text":"Richard P. Brent","submitted_at":"2016-08-17T02:12:14Z","abstract_excerpt":"We show that a well-known asymptotic series for the logarithm of the central binomial coefficient is strictly enveloping in the sense of P\\'olya and Szeg\\\"o, so the error incurred in truncating the series is of the same sign as the next term, and is bounded in magnitude by that term. We consider closely related asymptotic series for Binet's function, for $\\ln\\Gamma(z+1/2)$, and for the Riemann-Siegel theta function, and make some historical remarks."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.04834","kind":"arxiv","version":4},"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/1608.04834/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":"2026-07-05T03:41:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"X7NrN4eI9w4hcOlteP7S7fwBbWir3GrmTt+Vs3izSzbGZmB6QpMNip1Y6wx6IfhPZsUz44VDE6nLHmHLC5LTAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T18:47:03.524882Z"},"content_sha256":"4967e54c952bc74900e04123e10dcd983ac43bc44625ebd6a3e77f1990dffedd","schema_version":"1.0","event_id":"sha256:4967e54c952bc74900e04123e10dcd983ac43bc44625ebd6a3e77f1990dffedd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TOODYODFSJD5DVFEHYLXG4MTUA/bundle.json","state_url":"https://pith.science/pith/TOODYODFSJD5DVFEHYLXG4MTUA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TOODYODFSJD5DVFEHYLXG4MTUA/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":"2026-08-12T18:47:03Z","links":{"resolver":"https://pith.science/pith/TOODYODFSJD5DVFEHYLXG4MTUA","bundle":"https://pith.science/pith/TOODYODFSJD5DVFEHYLXG4MTUA/bundle.json","state":"https://pith.science/pith/TOODYODFSJD5DVFEHYLXG4MTUA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TOODYODFSJD5DVFEHYLXG4MTUA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:TOODYODFSJD5DVFEHYLXG4MTUA","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":"33b019b582a8594dfaec3104bf45313ac32dbead8ba1f3e7e15e68205b5691fe","cross_cats_sorted":["cs.NA","math.CA","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-08-17T02:12:14Z","title_canon_sha256":"948c3bc467ce950dd8eb073ae1194d806dda4c359b4da7aa011a7c9483f99906"},"schema_version":"1.0","source":{"id":"1608.04834","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.04834","created_at":"2026-07-05T03:41:55Z"},{"alias_kind":"arxiv_version","alias_value":"1608.04834v4","created_at":"2026-07-05T03:41:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.04834","created_at":"2026-07-05T03:41:55Z"},{"alias_kind":"pith_short_12","alias_value":"TOODYODFSJD5","created_at":"2026-07-05T03:41:55Z"},{"alias_kind":"pith_short_16","alias_value":"TOODYODFSJD5DVFE","created_at":"2026-07-05T03:41:55Z"},{"alias_kind":"pith_short_8","alias_value":"TOODYODF","created_at":"2026-07-05T03:41:55Z"}],"graph_snapshots":[{"event_id":"sha256:4967e54c952bc74900e04123e10dcd983ac43bc44625ebd6a3e77f1990dffedd","target":"graph","created_at":"2026-07-05T03:41:55Z","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/1608.04834/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We show that a well-known asymptotic series for the logarithm of the central binomial coefficient is strictly enveloping in the sense of P\\'olya and Szeg\\\"o, so the error incurred in truncating the series is of the same sign as the next term, and is bounded in magnitude by that term. We consider closely related asymptotic series for Binet's function, for $\\ln\\Gamma(z+1/2)$, and for the Riemann-Siegel theta function, and make some historical remarks.","authors_text":"Richard P. Brent","cross_cats":["cs.NA","math.CA","math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-08-17T02:12:14Z","title":"Asymptotic approximation of central binomial coefficients with rigorous error bounds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.04834","kind":"arxiv","version":4},"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":"sha256:64c15f77d97e7afca7a9e9645542620f221a6d99023651c2b682af91742b4417","target":"record","created_at":"2026-07-05T03:41:55Z","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":"33b019b582a8594dfaec3104bf45313ac32dbead8ba1f3e7e15e68205b5691fe","cross_cats_sorted":["cs.NA","math.CA","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-08-17T02:12:14Z","title_canon_sha256":"948c3bc467ce950dd8eb073ae1194d806dda4c359b4da7aa011a7c9483f99906"},"schema_version":"1.0","source":{"id":"1608.04834","kind":"arxiv","version":4}},"canonical_sha256":"9b9c3c38659247d1d4a43e17737193a03023bef7ab4d1e22171380fa68dcd87a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9b9c3c38659247d1d4a43e17737193a03023bef7ab4d1e22171380fa68dcd87a","first_computed_at":"2026-07-05T03:41:55.472538Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:41:55.472538Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/Znpjm06+QB9/lC+fFV4lT5GSUjmLDi0zrpyaU0wKwkXqj56sp8h3gCyn92y9XrrJyjZZOONBpbKDhXFZABGCg==","signature_status":"signed_v1","signed_at":"2026-07-05T03:41:55.472889Z","signed_message":"canonical_sha256_bytes"},"source_id":"1608.04834","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:64c15f77d97e7afca7a9e9645542620f221a6d99023651c2b682af91742b4417","sha256:4967e54c952bc74900e04123e10dcd983ac43bc44625ebd6a3e77f1990dffedd"],"state_sha256":"fe5b66dac1e8d97b20efd59586c09476c158f40248d5a0e05971f754272ce797"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"V7yX+1uFQa9ZhtewrH4CkPtiSgYK1E7GPGT6qdwhivwMKukvp7W4yheyoBfuwJwdwFb8QdvX5bjwx5+Y1fX9AA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T18:47:03.531506Z","bundle_sha256":"6fff4cb561f0ae8f687cdbd915726d87c73f5382b9dee8fd86bd6734317903a3"}}