{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:YIORBODR3SMZ37X665R7CP3AK6","short_pith_number":"pith:YIORBODR","canonical_record":{"source":{"id":"1512.00699","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-12-02T14:10:33Z","cross_cats_sorted":[],"title_canon_sha256":"f1a36987dd0d54d11798cf8b7e355d7adb6f5029e6bb6bd44882da3c4efc8bc5","abstract_canon_sha256":"c4302c7ac812ed9b245ee8721eb9010da42550fea29b420a279df72ed801da9b"},"schema_version":"1.0"},"canonical_sha256":"c21d10b871dc999dfefef763f13f6057b09f07cb4da1ad9a9380e93b96e0e853","source":{"kind":"arxiv","id":"1512.00699","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.00699","created_at":"2026-05-18T01:25:25Z"},{"alias_kind":"arxiv_version","alias_value":"1512.00699v1","created_at":"2026-05-18T01:25:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.00699","created_at":"2026-05-18T01:25:25Z"},{"alias_kind":"pith_short_12","alias_value":"YIORBODR3SMZ","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_16","alias_value":"YIORBODR3SMZ37X6","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_8","alias_value":"YIORBODR","created_at":"2026-05-18T12:29:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:YIORBODR3SMZ37X665R7CP3AK6","target":"record","payload":{"canonical_record":{"source":{"id":"1512.00699","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-12-02T14:10:33Z","cross_cats_sorted":[],"title_canon_sha256":"f1a36987dd0d54d11798cf8b7e355d7adb6f5029e6bb6bd44882da3c4efc8bc5","abstract_canon_sha256":"c4302c7ac812ed9b245ee8721eb9010da42550fea29b420a279df72ed801da9b"},"schema_version":"1.0"},"canonical_sha256":"c21d10b871dc999dfefef763f13f6057b09f07cb4da1ad9a9380e93b96e0e853","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:25:25.349733Z","signature_b64":"jsoTtWTzn9k1unlyQLjitsxwkW1gKkwyA0gCCXgPdtcpI80hrgZTzU+XOJzT9ADEgijvSPbKPHUtmXm14I6JDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c21d10b871dc999dfefef763f13f6057b09f07cb4da1ad9a9380e93b96e0e853","last_reissued_at":"2026-05-18T01:25:25.349209Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:25:25.349209Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1512.00699","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":"2026-05-18T01:25:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3cUb582gXVbgwWRqlnqsoKVspz7zBse8+HxSCkQNW4MX1AbJHK/WiBlgYPS0n5zm086IRUg5AWrXB27ll6GPCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T22:10:05.330972Z"},"content_sha256":"41ada70167bbbc5f3cdb723d545277b330df27a22a4d2a5f1358405af9cfd8bd","schema_version":"1.0","event_id":"sha256:41ada70167bbbc5f3cdb723d545277b330df27a22a4d2a5f1358405af9cfd8bd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:YIORBODR3SMZ37X665R7CP3AK6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Correction to Section 19.2 of Ricci Flow and the Poincare Conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Gang Tian, John Morgan","submitted_at":"2015-12-02T14:10:33Z","abstract_excerpt":"This note corrects a mistake in the original book in the evolution equations of total curvature for the curve-shrinking flow in an ambient Ricci Flow. The resulting upper bound for the evolution of total curvature is an exponential bound in time. The change involves the multiplicative constant. Here we show that it depends on the initial total curvature and the initial length, rather than just on the initial total curvature as was asserted before. This change does not affect the application of these results to prove finite-time extinction when the third homotopy group of the manifold is non-tr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.00699","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":"2026-05-18T01:25:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Js+HCauDM3j+7Gvv0cFHmUeR+VAZ7NQNvT5nSgpcBo9C0Jq+dQnDyCNOCTvq/bykwrgG/5HZxMYafnq21kyHBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T22:10:05.331393Z"},"content_sha256":"f615da3a283f10261832da9c6038f794215c87d8286ba485d271c9b3f69418e1","schema_version":"1.0","event_id":"sha256:f615da3a283f10261832da9c6038f794215c87d8286ba485d271c9b3f69418e1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YIORBODR3SMZ37X665R7CP3AK6/bundle.json","state_url":"https://pith.science/pith/YIORBODR3SMZ37X665R7CP3AK6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YIORBODR3SMZ37X665R7CP3AK6/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":"2026-05-28T22:10:05Z","links":{"resolver":"https://pith.science/pith/YIORBODR3SMZ37X665R7CP3AK6","bundle":"https://pith.science/pith/YIORBODR3SMZ37X665R7CP3AK6/bundle.json","state":"https://pith.science/pith/YIORBODR3SMZ37X665R7CP3AK6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YIORBODR3SMZ37X665R7CP3AK6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:YIORBODR3SMZ37X665R7CP3AK6","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":"c4302c7ac812ed9b245ee8721eb9010da42550fea29b420a279df72ed801da9b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-12-02T14:10:33Z","title_canon_sha256":"f1a36987dd0d54d11798cf8b7e355d7adb6f5029e6bb6bd44882da3c4efc8bc5"},"schema_version":"1.0","source":{"id":"1512.00699","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.00699","created_at":"2026-05-18T01:25:25Z"},{"alias_kind":"arxiv_version","alias_value":"1512.00699v1","created_at":"2026-05-18T01:25:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.00699","created_at":"2026-05-18T01:25:25Z"},{"alias_kind":"pith_short_12","alias_value":"YIORBODR3SMZ","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_16","alias_value":"YIORBODR3SMZ37X6","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_8","alias_value":"YIORBODR","created_at":"2026-05-18T12:29:50Z"}],"graph_snapshots":[{"event_id":"sha256:f615da3a283f10261832da9c6038f794215c87d8286ba485d271c9b3f69418e1","target":"graph","created_at":"2026-05-18T01:25:25Z","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":"This note corrects a mistake in the original book in the evolution equations of total curvature for the curve-shrinking flow in an ambient Ricci Flow. The resulting upper bound for the evolution of total curvature is an exponential bound in time. The change involves the multiplicative constant. Here we show that it depends on the initial total curvature and the initial length, rather than just on the initial total curvature as was asserted before. This change does not affect the application of these results to prove finite-time extinction when the third homotopy group of the manifold is non-tr","authors_text":"Gang Tian, John Morgan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-12-02T14:10:33Z","title":"Correction to Section 19.2 of Ricci Flow and the Poincare Conjecture"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.00699","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":"sha256:41ada70167bbbc5f3cdb723d545277b330df27a22a4d2a5f1358405af9cfd8bd","target":"record","created_at":"2026-05-18T01:25:25Z","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":"c4302c7ac812ed9b245ee8721eb9010da42550fea29b420a279df72ed801da9b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-12-02T14:10:33Z","title_canon_sha256":"f1a36987dd0d54d11798cf8b7e355d7adb6f5029e6bb6bd44882da3c4efc8bc5"},"schema_version":"1.0","source":{"id":"1512.00699","kind":"arxiv","version":1}},"canonical_sha256":"c21d10b871dc999dfefef763f13f6057b09f07cb4da1ad9a9380e93b96e0e853","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c21d10b871dc999dfefef763f13f6057b09f07cb4da1ad9a9380e93b96e0e853","first_computed_at":"2026-05-18T01:25:25.349209Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:25:25.349209Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jsoTtWTzn9k1unlyQLjitsxwkW1gKkwyA0gCCXgPdtcpI80hrgZTzU+XOJzT9ADEgijvSPbKPHUtmXm14I6JDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:25:25.349733Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.00699","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:41ada70167bbbc5f3cdb723d545277b330df27a22a4d2a5f1358405af9cfd8bd","sha256:f615da3a283f10261832da9c6038f794215c87d8286ba485d271c9b3f69418e1"],"state_sha256":"bbc8ea5680b6190348c1167c6945dbf472849b8ba2ee0d87317261a8b475d9c7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qjgApMAyTHtrTHP+rhcddfsVczhe3eamwfQ8Iv2lD/X2lGYZjfOxDSYQ67JQQgpPkev7lnCjJrsRxjjKJJAfBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T22:10:05.335120Z","bundle_sha256":"3052839263cc36bbbf7bef9d2932f589f1bbc1a4fe549bb906166fe567d16985"}}