{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:4FP6FVE7GLMUOZFENXOX4GZ4MM","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":"74986f0d4b1503846bcc96c0e75f8419eb2e318a2147c1f4d0bfaa44f6fe03a9","cross_cats_sorted":["math-ph","math.DG","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2012-01-20T10:22:14Z","title_canon_sha256":"85933dc92eae3f0d198b62d2f2e3693396e33f002de836eff82af6973e5c09b9"},"schema_version":"1.0","source":{"id":"1201.4249","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.4249","created_at":"2026-05-18T03:30:42Z"},{"alias_kind":"arxiv_version","alias_value":"1201.4249v1","created_at":"2026-05-18T03:30:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.4249","created_at":"2026-05-18T03:30:42Z"},{"alias_kind":"pith_short_12","alias_value":"4FP6FVE7GLMU","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"4FP6FVE7GLMUOZFE","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"4FP6FVE7","created_at":"2026-05-18T12:26:53Z"}],"graph_snapshots":[{"event_id":"sha256:a58f00a75dd3400e0b4c41fcd4563750cfe383f2a9c32abdd66d8e29798ae722","target":"graph","created_at":"2026-05-18T03:30:42Z","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 develop area and volume comparison theorems for the evolution of spacelike, acausal, causally complete hypersurfaces in Lorentzian manifolds, where one has a lower bound on the Ricci tensor along timelike curves, and an upper bound on the mean curvature of the hypersurface. Using these results, we give a new proof of Hawking's singularity theorem.","authors_text":"James D.E. Grant, Jan-Hendrik Treude","cross_cats":["math-ph","math.DG","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2012-01-20T10:22:14Z","title":"Volume Comparison for Hypersurfaces in Lorentzian Manifolds and Singularity Theorems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.4249","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":"sha256:57d885235ac583ef5103aefc9e2f15d1e6d7c1eb3e4b93fdac9bac58d35f81dd","target":"record","created_at":"2026-05-18T03:30:42Z","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":"74986f0d4b1503846bcc96c0e75f8419eb2e318a2147c1f4d0bfaa44f6fe03a9","cross_cats_sorted":["math-ph","math.DG","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"gr-qc","submitted_at":"2012-01-20T10:22:14Z","title_canon_sha256":"85933dc92eae3f0d198b62d2f2e3693396e33f002de836eff82af6973e5c09b9"},"schema_version":"1.0","source":{"id":"1201.4249","kind":"arxiv","version":1}},"canonical_sha256":"e15fe2d49f32d94764a46ddd7e1b3c632971907a3358abc1f7fc42ee4ca8f66e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e15fe2d49f32d94764a46ddd7e1b3c632971907a3358abc1f7fc42ee4ca8f66e","first_computed_at":"2026-05-18T03:30:42.314641Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:30:42.314641Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"y4Wg1kZdIZUAOFqrUJYI5QmpVaejWyUokIO2Vu3Kmp34RVWGlhXtY4XXJL/sc4sNAZV/xDY2jW9JEeUs2dJNAg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:30:42.315485Z","signed_message":"canonical_sha256_bytes"},"source_id":"1201.4249","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:57d885235ac583ef5103aefc9e2f15d1e6d7c1eb3e4b93fdac9bac58d35f81dd","sha256:a58f00a75dd3400e0b4c41fcd4563750cfe383f2a9c32abdd66d8e29798ae722"],"state_sha256":"7d7e98d0c32cc906c539a5039b875deec19a58f42091dd10e68c693c288bba9e"}