{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:OQQHFD5253QUJZKI2F3TSBMTBX","short_pith_number":"pith:OQQHFD52","canonical_record":{"source":{"id":"2108.11852","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2021-08-26T15:23:48Z","cross_cats_sorted":[],"title_canon_sha256":"21f6e44043ad10c45549a25dc2c461b22363649565dd0c2ad7841898d7c5bb5b","abstract_canon_sha256":"72d7ca1904f039d04a3505404956b4b8584687638ee6d277d0fd417366b1d1c4"},"schema_version":"1.0"},"canonical_sha256":"7420728fbaeee144e548d1773905930dc386ec89ec6c7564b41ad956ad9b0410","source":{"kind":"arxiv","id":"2108.11852","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2108.11852","created_at":"2026-07-05T03:09:10Z"},{"alias_kind":"arxiv_version","alias_value":"2108.11852v1","created_at":"2026-07-05T03:09:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2108.11852","created_at":"2026-07-05T03:09:10Z"},{"alias_kind":"pith_short_12","alias_value":"OQQHFD5253QU","created_at":"2026-07-05T03:09:10Z"},{"alias_kind":"pith_short_16","alias_value":"OQQHFD5253QUJZKI","created_at":"2026-07-05T03:09:10Z"},{"alias_kind":"pith_short_8","alias_value":"OQQHFD52","created_at":"2026-07-05T03:09:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:OQQHFD5253QUJZKI2F3TSBMTBX","target":"record","payload":{"canonical_record":{"source":{"id":"2108.11852","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2021-08-26T15:23:48Z","cross_cats_sorted":[],"title_canon_sha256":"21f6e44043ad10c45549a25dc2c461b22363649565dd0c2ad7841898d7c5bb5b","abstract_canon_sha256":"72d7ca1904f039d04a3505404956b4b8584687638ee6d277d0fd417366b1d1c4"},"schema_version":"1.0"},"canonical_sha256":"7420728fbaeee144e548d1773905930dc386ec89ec6c7564b41ad956ad9b0410","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:09:10.375584Z","signature_b64":"n9z/xWNWNTYVg1EM1i6kmJcUQNH1FhB1MFvrh86VS89fH8YZV3yN4bQhrFRiREmB7FJC198J2i8w1D2T/6WXCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7420728fbaeee144e548d1773905930dc386ec89ec6c7564b41ad956ad9b0410","last_reissued_at":"2026-07-05T03:09:10.375163Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:09:10.375163Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2108.11852","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":"2026-07-05T03:09:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kpxa4N2OKjbXknFEV2tq3RQq4hQn8f8J56tIlGWwRAnxZWckAXJnSEIfQZVCfCbFEB65CKUxjVwZrfRTLkw/DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T10:44:31.220577Z"},"content_sha256":"c236ecdabd65f56b0ae23aaa816ed3978fafdb55047b56d9c9119f43b49f0d9e","schema_version":"1.0","event_id":"sha256:c236ecdabd65f56b0ae23aaa816ed3978fafdb55047b56d9c9119f43b49f0d9e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:OQQHFD5253QUJZKI2F3TSBMTBX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Quantitative long range curvature estimate for mean curvature flow","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Jingze Zhu","submitted_at":"2021-08-26T15:23:48Z","abstract_excerpt":"We prove that smooth convex $\\alpha$-noncollapsed ancient mean curvature flow satisfies a quantitative curvature estimate $H(y,t)\\leq CH(x,t)(H(x,t)|x-y|+1)^2$ for any pair of $x,y$. In other words, the rescaled curvature grows at most quadratically in terms of the rescaled extrinsic distance."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2108.11852","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/2108.11852/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":"2026-07-05T03:09:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1+0D/BQ2jKHXnUlmbzoM8dCVX7j/YlYoXvmbOvyxDsuAHIpsnWUDsEMZJquSd1YI03q7ZuvOZOmPlnMV8Y9nCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T10:44:31.220940Z"},"content_sha256":"000e4e53dcf9a768477fa68f6e7935ecda6259efb297a8969ebe611d55b77dc8","schema_version":"1.0","event_id":"sha256:000e4e53dcf9a768477fa68f6e7935ecda6259efb297a8969ebe611d55b77dc8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OQQHFD5253QUJZKI2F3TSBMTBX/bundle.json","state_url":"https://pith.science/pith/OQQHFD5253QUJZKI2F3TSBMTBX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OQQHFD5253QUJZKI2F3TSBMTBX/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":"2026-07-06T10:44:31Z","links":{"resolver":"https://pith.science/pith/OQQHFD5253QUJZKI2F3TSBMTBX","bundle":"https://pith.science/pith/OQQHFD5253QUJZKI2F3TSBMTBX/bundle.json","state":"https://pith.science/pith/OQQHFD5253QUJZKI2F3TSBMTBX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OQQHFD5253QUJZKI2F3TSBMTBX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:OQQHFD5253QUJZKI2F3TSBMTBX","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":"72d7ca1904f039d04a3505404956b4b8584687638ee6d277d0fd417366b1d1c4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2021-08-26T15:23:48Z","title_canon_sha256":"21f6e44043ad10c45549a25dc2c461b22363649565dd0c2ad7841898d7c5bb5b"},"schema_version":"1.0","source":{"id":"2108.11852","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2108.11852","created_at":"2026-07-05T03:09:10Z"},{"alias_kind":"arxiv_version","alias_value":"2108.11852v1","created_at":"2026-07-05T03:09:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2108.11852","created_at":"2026-07-05T03:09:10Z"},{"alias_kind":"pith_short_12","alias_value":"OQQHFD5253QU","created_at":"2026-07-05T03:09:10Z"},{"alias_kind":"pith_short_16","alias_value":"OQQHFD5253QUJZKI","created_at":"2026-07-05T03:09:10Z"},{"alias_kind":"pith_short_8","alias_value":"OQQHFD52","created_at":"2026-07-05T03:09:10Z"}],"graph_snapshots":[{"event_id":"sha256:000e4e53dcf9a768477fa68f6e7935ecda6259efb297a8969ebe611d55b77dc8","target":"graph","created_at":"2026-07-05T03:09:10Z","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/2108.11852/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove that smooth convex $\\alpha$-noncollapsed ancient mean curvature flow satisfies a quantitative curvature estimate $H(y,t)\\leq CH(x,t)(H(x,t)|x-y|+1)^2$ for any pair of $x,y$. In other words, the rescaled curvature grows at most quadratically in terms of the rescaled extrinsic distance.","authors_text":"Jingze Zhu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2021-08-26T15:23:48Z","title":"Quantitative long range curvature estimate for mean curvature flow"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2108.11852","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:c236ecdabd65f56b0ae23aaa816ed3978fafdb55047b56d9c9119f43b49f0d9e","target":"record","created_at":"2026-07-05T03:09:10Z","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":"72d7ca1904f039d04a3505404956b4b8584687638ee6d277d0fd417366b1d1c4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2021-08-26T15:23:48Z","title_canon_sha256":"21f6e44043ad10c45549a25dc2c461b22363649565dd0c2ad7841898d7c5bb5b"},"schema_version":"1.0","source":{"id":"2108.11852","kind":"arxiv","version":1}},"canonical_sha256":"7420728fbaeee144e548d1773905930dc386ec89ec6c7564b41ad956ad9b0410","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7420728fbaeee144e548d1773905930dc386ec89ec6c7564b41ad956ad9b0410","first_computed_at":"2026-07-05T03:09:10.375163Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:09:10.375163Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"n9z/xWNWNTYVg1EM1i6kmJcUQNH1FhB1MFvrh86VS89fH8YZV3yN4bQhrFRiREmB7FJC198J2i8w1D2T/6WXCw==","signature_status":"signed_v1","signed_at":"2026-07-05T03:09:10.375584Z","signed_message":"canonical_sha256_bytes"},"source_id":"2108.11852","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c236ecdabd65f56b0ae23aaa816ed3978fafdb55047b56d9c9119f43b49f0d9e","sha256:000e4e53dcf9a768477fa68f6e7935ecda6259efb297a8969ebe611d55b77dc8"],"state_sha256":"a5d720e587d6786a60ff068e708c2e1dcaa925c8a3d3719b77f9be9881d2ea7a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UVpvRXoJLZvHTtdj+IOC2LXIri7SIQ9aYeHj+0j3DbqKxKrL13Ij96nx/0R8YmJB8GIPFwBOgNLrlS07CnOqBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-06T10:44:31.222892Z","bundle_sha256":"c0344aef105c1395fc349cf6405d7af23150018b4d459dbedb7a44106d223704"}}