{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:HSYJ43J5GG52VDLLN2RW5X5LUD","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":"70aff190adf2edfd7a7e712a8cf9d35167bb02ad7e2cc2b6d7aba01e027ce565","cross_cats_sorted":["gr-qc"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.DG","submitted_at":"2021-09-18T09:15:37Z","title_canon_sha256":"93e473574ee8f2a5fe922660c586ce53da5511bb97a1cc7837d1d7b736c0697c"},"schema_version":"1.0","source":{"id":"2109.08888","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2109.08888","created_at":"2026-07-05T05:42:22Z"},{"alias_kind":"arxiv_version","alias_value":"2109.08888v4","created_at":"2026-07-05T05:42:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2109.08888","created_at":"2026-07-05T05:42:22Z"},{"alias_kind":"pith_short_12","alias_value":"HSYJ43J5GG52","created_at":"2026-07-05T05:42:22Z"},{"alias_kind":"pith_short_16","alias_value":"HSYJ43J5GG52VDLL","created_at":"2026-07-05T05:42:22Z"},{"alias_kind":"pith_short_8","alias_value":"HSYJ43J5","created_at":"2026-07-05T05:42:22Z"}],"graph_snapshots":[{"event_id":"sha256:7ee7a6ada13f9544c5ca34d2e0f2fb716d41d9e2408c9064612decafd43f8c68","target":"graph","created_at":"2026-07-05T05:42:22Z","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/2109.08888/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we introduce the notion of a marginal tube, which is a hypersurface foliated by marginal surfaces. It generalises the notion of a marginally trapped tube and several notions of black hole horizons, for example trapping horizons, isolated horizons, dynamical horizons, etc. We prove that if every spacelike section of a marginal tube is a marginal surface, then the marginal tube is null. There is no assumption on the topology of the marginal tube. To prove it, we study the geometry of spacelike surfaces in a 4-dimensional spacetime with the help of double null coordinate systems. T","authors_text":"Pengyu Le","cross_cats":["gr-qc"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.DG","submitted_at":"2021-09-18T09:15:37Z","title":"Marginal tubes and foliations by marginal surfaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.08888","kind":"arxiv","version":4},"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":"sha256:3e2792e385665327efca506f4842f7c5e3d16f0c3fbad48592b88f5ab2351b2f","target":"record","created_at":"2026-07-05T05:42:22Z","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":"70aff190adf2edfd7a7e712a8cf9d35167bb02ad7e2cc2b6d7aba01e027ce565","cross_cats_sorted":["gr-qc"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.DG","submitted_at":"2021-09-18T09:15:37Z","title_canon_sha256":"93e473574ee8f2a5fe922660c586ce53da5511bb97a1cc7837d1d7b736c0697c"},"schema_version":"1.0","source":{"id":"2109.08888","kind":"arxiv","version":4}},"canonical_sha256":"3cb09e6d3d31bbaa8d6b6ea36edfaba0f8294f5008fd4c267a8cdb8a33084d7a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3cb09e6d3d31bbaa8d6b6ea36edfaba0f8294f5008fd4c267a8cdb8a33084d7a","first_computed_at":"2026-07-05T05:42:22.068655Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:42:22.068655Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ioFM1iqh4A2um3tyr+XWaIDqkxK+51HrLkEgoPcfNwZJp5OLM+pn5Pp3PL8gcIYJEhYHOYSRdMjB0xPfu2o1BQ==","signature_status":"signed_v1","signed_at":"2026-07-05T05:42:22.069155Z","signed_message":"canonical_sha256_bytes"},"source_id":"2109.08888","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3e2792e385665327efca506f4842f7c5e3d16f0c3fbad48592b88f5ab2351b2f","sha256:7ee7a6ada13f9544c5ca34d2e0f2fb716d41d9e2408c9064612decafd43f8c68"],"state_sha256":"d6f6c589b438e2b7bddb33cb77686b7de91dc1855ea2d3896296943a9d246a66"}