{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:5JUGKVLBGHNK3SSTYVLBUQSXMR","short_pith_number":"pith:5JUGKVLB","canonical_record":{"source":{"id":"1008.1605","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-08-09T23:43:46Z","cross_cats_sorted":["gr-qc"],"title_canon_sha256":"f725871425171c94da7ecaead08a1e7764f3def7bab45f2c4707f3ad14ee4491","abstract_canon_sha256":"43d38d21a61c67bde472eabbd8dc0947c49798ece5142845b2bc41045262e3ac"},"schema_version":"1.0"},"canonical_sha256":"ea6865556131daadca53c5561a425764773f0f53dcb5d8c4b14d491a78048017","source":{"kind":"arxiv","id":"1008.1605","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.1605","created_at":"2026-05-18T01:04:46Z"},{"alias_kind":"arxiv_version","alias_value":"1008.1605v2","created_at":"2026-05-18T01:04:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.1605","created_at":"2026-05-18T01:04:46Z"},{"alias_kind":"pith_short_12","alias_value":"5JUGKVLBGHNK","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_16","alias_value":"5JUGKVLBGHNK3SST","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_8","alias_value":"5JUGKVLB","created_at":"2026-05-18T12:26:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:5JUGKVLBGHNK3SSTYVLBUQSXMR","target":"record","payload":{"canonical_record":{"source":{"id":"1008.1605","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-08-09T23:43:46Z","cross_cats_sorted":["gr-qc"],"title_canon_sha256":"f725871425171c94da7ecaead08a1e7764f3def7bab45f2c4707f3ad14ee4491","abstract_canon_sha256":"43d38d21a61c67bde472eabbd8dc0947c49798ece5142845b2bc41045262e3ac"},"schema_version":"1.0"},"canonical_sha256":"ea6865556131daadca53c5561a425764773f0f53dcb5d8c4b14d491a78048017","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:46.210838Z","signature_b64":"irLMY4M33QJxFo3xF06/xBGCBgDM8FHgBrWMze+zG5sZDmUwYzbyco0Nf3pmKFnzqiqwTFE44baAMwHPO675Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ea6865556131daadca53c5561a425764773f0f53dcb5d8c4b14d491a78048017","last_reissued_at":"2026-05-18T01:04:46.210417Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:46.210417Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1008.1605","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":"2026-05-18T01:04:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KIfmiE0yTH7uxiaU3pR2LHeRwHPiKkvw396Ie4/USxVOyEfVhXLBBYLVZ4iojkxTW06M8b3tyvy5NVZIKGc2Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-20T14:19:32.442194Z"},"content_sha256":"648264f8deb662ae9412026521759b5da435ddac955bde831a830790ebfac19d","schema_version":"1.0","event_id":"sha256:648264f8deb662ae9412026521759b5da435ddac955bde831a830790ebfac19d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:5JUGKVLBGHNK3SSTYVLBUQSXMR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Breakdown Criteria for Nonvacuum Einstein Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["gr-qc"],"primary_cat":"math.AP","authors_text":"Arick Shao","submitted_at":"2010-08-09T23:43:46Z","abstract_excerpt":"The recent \"breakdown criterion\" result of S. Klainerman and I. Rodnianski stated roughly that an Einstein-vacuum spacetime, given as a CMC foliation, can be further extended in time if the second fundamental form and the derivative of the lapse of the foliation are uniformly bounded. This theorem and its proof were extended to Einstein-scalar and Einstein-Maxwell spacetimes in the author's Ph.D. thesis. In this paper, we state the main results of the thesis, and we summarize and discuss their proofs. In particular, we will discuss the various issues resulting from nontrivial Ricci curvature a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.1605","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":"2026-05-18T01:04:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"elX6w2E9ji7f/bdyNQaudls07EtedlFp4S6JmnvDYV0wgCj7ZolAJKP0lGUzQhNjydT58pprRYGsWBY+0PNRAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-20T14:19:32.442555Z"},"content_sha256":"971ff497120a66ac2bdc7e7fa848c63c40707b75c9c013f14a4ffea458941091","schema_version":"1.0","event_id":"sha256:971ff497120a66ac2bdc7e7fa848c63c40707b75c9c013f14a4ffea458941091"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5JUGKVLBGHNK3SSTYVLBUQSXMR/bundle.json","state_url":"https://pith.science/pith/5JUGKVLBGHNK3SSTYVLBUQSXMR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5JUGKVLBGHNK3SSTYVLBUQSXMR/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":"2026-05-20T14:19:32Z","links":{"resolver":"https://pith.science/pith/5JUGKVLBGHNK3SSTYVLBUQSXMR","bundle":"https://pith.science/pith/5JUGKVLBGHNK3SSTYVLBUQSXMR/bundle.json","state":"https://pith.science/pith/5JUGKVLBGHNK3SSTYVLBUQSXMR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5JUGKVLBGHNK3SSTYVLBUQSXMR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:5JUGKVLBGHNK3SSTYVLBUQSXMR","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":"43d38d21a61c67bde472eabbd8dc0947c49798ece5142845b2bc41045262e3ac","cross_cats_sorted":["gr-qc"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-08-09T23:43:46Z","title_canon_sha256":"f725871425171c94da7ecaead08a1e7764f3def7bab45f2c4707f3ad14ee4491"},"schema_version":"1.0","source":{"id":"1008.1605","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.1605","created_at":"2026-05-18T01:04:46Z"},{"alias_kind":"arxiv_version","alias_value":"1008.1605v2","created_at":"2026-05-18T01:04:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.1605","created_at":"2026-05-18T01:04:46Z"},{"alias_kind":"pith_short_12","alias_value":"5JUGKVLBGHNK","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_16","alias_value":"5JUGKVLBGHNK3SST","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_8","alias_value":"5JUGKVLB","created_at":"2026-05-18T12:26:04Z"}],"graph_snapshots":[{"event_id":"sha256:971ff497120a66ac2bdc7e7fa848c63c40707b75c9c013f14a4ffea458941091","target":"graph","created_at":"2026-05-18T01:04:46Z","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":"The recent \"breakdown criterion\" result of S. Klainerman and I. Rodnianski stated roughly that an Einstein-vacuum spacetime, given as a CMC foliation, can be further extended in time if the second fundamental form and the derivative of the lapse of the foliation are uniformly bounded. This theorem and its proof were extended to Einstein-scalar and Einstein-Maxwell spacetimes in the author's Ph.D. thesis. In this paper, we state the main results of the thesis, and we summarize and discuss their proofs. In particular, we will discuss the various issues resulting from nontrivial Ricci curvature a","authors_text":"Arick Shao","cross_cats":["gr-qc"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-08-09T23:43:46Z","title":"On Breakdown Criteria for Nonvacuum Einstein Equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.1605","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":"sha256:648264f8deb662ae9412026521759b5da435ddac955bde831a830790ebfac19d","target":"record","created_at":"2026-05-18T01:04:46Z","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":"43d38d21a61c67bde472eabbd8dc0947c49798ece5142845b2bc41045262e3ac","cross_cats_sorted":["gr-qc"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-08-09T23:43:46Z","title_canon_sha256":"f725871425171c94da7ecaead08a1e7764f3def7bab45f2c4707f3ad14ee4491"},"schema_version":"1.0","source":{"id":"1008.1605","kind":"arxiv","version":2}},"canonical_sha256":"ea6865556131daadca53c5561a425764773f0f53dcb5d8c4b14d491a78048017","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ea6865556131daadca53c5561a425764773f0f53dcb5d8c4b14d491a78048017","first_computed_at":"2026-05-18T01:04:46.210417Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:04:46.210417Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"irLMY4M33QJxFo3xF06/xBGCBgDM8FHgBrWMze+zG5sZDmUwYzbyco0Nf3pmKFnzqiqwTFE44baAMwHPO675Bw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:04:46.210838Z","signed_message":"canonical_sha256_bytes"},"source_id":"1008.1605","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:648264f8deb662ae9412026521759b5da435ddac955bde831a830790ebfac19d","sha256:971ff497120a66ac2bdc7e7fa848c63c40707b75c9c013f14a4ffea458941091"],"state_sha256":"35250d13fc4a819327797167212047af68f04c3e12bc8adeca2b24548af2a648"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FEFVGar67mxCKPTEScb+EtCfCI/M6jj0Gw4/0NKyeqWuqord5DfHma1X10IQWE7bbMecfxCtRmmBTOGtn9ynCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-20T14:19:32.444408Z","bundle_sha256":"b2bc8c715ddd12790eb66d04d4755685475c59e04ca56ff5d5ed6ee99839525e"}}