{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:K5YGYCGIDJGPCXAWMFOBLDCQQ7","short_pith_number":"pith:K5YGYCGI","canonical_record":{"source":{"id":"1212.1428","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-11-27T17:07:49Z","cross_cats_sorted":[],"title_canon_sha256":"a2abd004ef8b7c6081100a399aa1c562eb7c84d4b95dd33765c35029763624bf","abstract_canon_sha256":"d69aba5e0416736503a0046166c2870c5102fefe6871c44d404de32a05222ba9"},"schema_version":"1.0"},"canonical_sha256":"57706c08c81a4cf15c16615c158c5087e94dcbb98f6d568d3da991c7f57123a4","source":{"kind":"arxiv","id":"1212.1428","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.1428","created_at":"2026-05-18T03:39:03Z"},{"alias_kind":"arxiv_version","alias_value":"1212.1428v1","created_at":"2026-05-18T03:39:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.1428","created_at":"2026-05-18T03:39:03Z"},{"alias_kind":"pith_short_12","alias_value":"K5YGYCGIDJGP","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"K5YGYCGIDJGPCXAW","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"K5YGYCGI","created_at":"2026-05-18T12:27:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:K5YGYCGIDJGPCXAWMFOBLDCQQ7","target":"record","payload":{"canonical_record":{"source":{"id":"1212.1428","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-11-27T17:07:49Z","cross_cats_sorted":[],"title_canon_sha256":"a2abd004ef8b7c6081100a399aa1c562eb7c84d4b95dd33765c35029763624bf","abstract_canon_sha256":"d69aba5e0416736503a0046166c2870c5102fefe6871c44d404de32a05222ba9"},"schema_version":"1.0"},"canonical_sha256":"57706c08c81a4cf15c16615c158c5087e94dcbb98f6d568d3da991c7f57123a4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:39:03.977354Z","signature_b64":"ECPKU+GVH5/3Vu2Dqy4x9aDYMqmqJH1axPtb+PMnb++ujcValdG3PQaw+F9PW/ND033EpgYVqK1vr9HN6lEuDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"57706c08c81a4cf15c16615c158c5087e94dcbb98f6d568d3da991c7f57123a4","last_reissued_at":"2026-05-18T03:39:03.976917Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:39:03.976917Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1212.1428","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":"2026-05-18T03:39:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NsWJM6gTvxMBYfFd98uv9gVHb00PtfYjMkswv4ovTRtb3hgdP4pkG2I4WPDocERQkd2Nf3Vf4F0xA4QAggvODA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T14:56:33.431284Z"},"content_sha256":"09aa0f62197f345112d05986da04fef19019a046acaae0573728cd839937d5da","schema_version":"1.0","event_id":"sha256:09aa0f62197f345112d05986da04fef19019a046acaae0573728cd839937d5da"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:K5YGYCGIDJGPCXAWMFOBLDCQQ7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Refinement of Ramanujan's Factorial Approximation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Mark B. Villarino, Michael D. Hirschhorn","submitted_at":"2012-11-27T17:07:49Z","abstract_excerpt":"We give an elementary proof for new strict upper and lower bounds for the correction term in Ramanujan's approximation for the factorial function"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.1428","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":""},"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":"2026-05-18T03:39:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OG9mcYqJaEuG+cESvh8abvOACZSwObWudmdHhp9h2azx8bMz6EYn/FbVG8ZAwyMHvXjl4auSXXAlto12WHPDCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T14:56:33.431771Z"},"content_sha256":"e8cca0446381432a51b9e41d4240b0e09e3c31aef1a8d1f53ea9a81028ba3f86","schema_version":"1.0","event_id":"sha256:e8cca0446381432a51b9e41d4240b0e09e3c31aef1a8d1f53ea9a81028ba3f86"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/K5YGYCGIDJGPCXAWMFOBLDCQQ7/bundle.json","state_url":"https://pith.science/pith/K5YGYCGIDJGPCXAWMFOBLDCQQ7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/K5YGYCGIDJGPCXAWMFOBLDCQQ7/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":"2026-05-27T14:56:33Z","links":{"resolver":"https://pith.science/pith/K5YGYCGIDJGPCXAWMFOBLDCQQ7","bundle":"https://pith.science/pith/K5YGYCGIDJGPCXAWMFOBLDCQQ7/bundle.json","state":"https://pith.science/pith/K5YGYCGIDJGPCXAWMFOBLDCQQ7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/K5YGYCGIDJGPCXAWMFOBLDCQQ7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:K5YGYCGIDJGPCXAWMFOBLDCQQ7","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":"d69aba5e0416736503a0046166c2870c5102fefe6871c44d404de32a05222ba9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-11-27T17:07:49Z","title_canon_sha256":"a2abd004ef8b7c6081100a399aa1c562eb7c84d4b95dd33765c35029763624bf"},"schema_version":"1.0","source":{"id":"1212.1428","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.1428","created_at":"2026-05-18T03:39:03Z"},{"alias_kind":"arxiv_version","alias_value":"1212.1428v1","created_at":"2026-05-18T03:39:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.1428","created_at":"2026-05-18T03:39:03Z"},{"alias_kind":"pith_short_12","alias_value":"K5YGYCGIDJGP","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"K5YGYCGIDJGPCXAW","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"K5YGYCGI","created_at":"2026-05-18T12:27:11Z"}],"graph_snapshots":[{"event_id":"sha256:e8cca0446381432a51b9e41d4240b0e09e3c31aef1a8d1f53ea9a81028ba3f86","target":"graph","created_at":"2026-05-18T03:39:03Z","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 give an elementary proof for new strict upper and lower bounds for the correction term in Ramanujan's approximation for the factorial function","authors_text":"Mark B. Villarino, Michael D. Hirschhorn","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-11-27T17:07:49Z","title":"A Refinement of Ramanujan's Factorial Approximation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.1428","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":"sha256:09aa0f62197f345112d05986da04fef19019a046acaae0573728cd839937d5da","target":"record","created_at":"2026-05-18T03:39:03Z","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":"d69aba5e0416736503a0046166c2870c5102fefe6871c44d404de32a05222ba9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-11-27T17:07:49Z","title_canon_sha256":"a2abd004ef8b7c6081100a399aa1c562eb7c84d4b95dd33765c35029763624bf"},"schema_version":"1.0","source":{"id":"1212.1428","kind":"arxiv","version":1}},"canonical_sha256":"57706c08c81a4cf15c16615c158c5087e94dcbb98f6d568d3da991c7f57123a4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"57706c08c81a4cf15c16615c158c5087e94dcbb98f6d568d3da991c7f57123a4","first_computed_at":"2026-05-18T03:39:03.976917Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:39:03.976917Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ECPKU+GVH5/3Vu2Dqy4x9aDYMqmqJH1axPtb+PMnb++ujcValdG3PQaw+F9PW/ND033EpgYVqK1vr9HN6lEuDw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:39:03.977354Z","signed_message":"canonical_sha256_bytes"},"source_id":"1212.1428","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:09aa0f62197f345112d05986da04fef19019a046acaae0573728cd839937d5da","sha256:e8cca0446381432a51b9e41d4240b0e09e3c31aef1a8d1f53ea9a81028ba3f86"],"state_sha256":"cafde736294a7bda36ddc53b84dc1ef75b14de973d2cbb3de874061513bc2b00"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HtkMO62TJLkR4y1tWsQUTaUhxFYLU9FFbuqyRki16g0lxvaePH2H1xrT5eiVxMs7c5/RFiSggTswRtqhdyySDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T14:56:33.434245Z","bundle_sha256":"57cec51748964f07cfd14e5d997605ccd9dae64698130c70c94595a3a71290e3"}}