{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:4MCGDEXWGX3XVO4E6X5JCBEQOB","short_pith_number":"pith:4MCGDEXW","canonical_record":{"source":{"id":"1711.02261","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-11-07T02:23:46Z","cross_cats_sorted":[],"title_canon_sha256":"6bc908f07e0469d6f533d4cdc015d66eb83fc755a3e4afd2cea39fbbe93d0012","abstract_canon_sha256":"98608f1192791812afbae8ebfafa7a9d3ea5103e7a9b049af11ab8bdcc53c727"},"schema_version":"1.0"},"canonical_sha256":"e3046192f635f77abb84f5fa910490706b82ec582586606f7c1f9682280c2b1a","source":{"kind":"arxiv","id":"1711.02261","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1711.02261","created_at":"2026-05-18T00:31:08Z"},{"alias_kind":"arxiv_version","alias_value":"1711.02261v1","created_at":"2026-05-18T00:31:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.02261","created_at":"2026-05-18T00:31:08Z"},{"alias_kind":"pith_short_12","alias_value":"4MCGDEXWGX3X","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_16","alias_value":"4MCGDEXWGX3XVO4E","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_8","alias_value":"4MCGDEXW","created_at":"2026-05-18T12:31:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:4MCGDEXWGX3XVO4E6X5JCBEQOB","target":"record","payload":{"canonical_record":{"source":{"id":"1711.02261","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-11-07T02:23:46Z","cross_cats_sorted":[],"title_canon_sha256":"6bc908f07e0469d6f533d4cdc015d66eb83fc755a3e4afd2cea39fbbe93d0012","abstract_canon_sha256":"98608f1192791812afbae8ebfafa7a9d3ea5103e7a9b049af11ab8bdcc53c727"},"schema_version":"1.0"},"canonical_sha256":"e3046192f635f77abb84f5fa910490706b82ec582586606f7c1f9682280c2b1a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:31:08.397844Z","signature_b64":"IkywI4Xqhh7Re2pc/JPNEIXHpbexX1AUx0PcR/8Xi1SKql0EPd9JsuZSvUuxmQF/6VhpQC46hlfcPL4ApiJ7Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e3046192f635f77abb84f5fa910490706b82ec582586606f7c1f9682280c2b1a","last_reissued_at":"2026-05-18T00:31:08.397163Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:31:08.397163Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1711.02261","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":"2026-05-18T00:31:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Kk2XZ//dGFpT5e2NRiOVMSCFlVz8OZttW8aKSMyEt3ufpSWLOLWTw7Ur6/EDcou4+LDbDPHoX9JmeHc5Qpu5Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T07:26:01.484220Z"},"content_sha256":"bb7cc2906271b2027a1a8d80d077fee6cbdb402c71e8543de4eb681c299c0d24","schema_version":"1.0","event_id":"sha256:bb7cc2906271b2027a1a8d80d077fee6cbdb402c71e8543de4eb681c299c0d24"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:4MCGDEXWGX3XVO4E6X5JCBEQOB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Liouville Theorem for Mean Curvature Flow","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Kevin Sonnanburg","submitted_at":"2017-11-07T02:23:46Z","abstract_excerpt":"Ancient solutions arise in the study of parabolic blow-ups. If we can categorize ancient solutions, we can better understand blow-up limits. Based on an argument of Giga and Kohn, we give a Liouville-type theorem restricting ancient, type-I, non-collapsing two- dimensional mean curvature flows to either spheres or cylinders."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.02261","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":"2026-05-18T00:31:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MAxN1f1R8n1IIQQA3WpbVisahqAYKY7Ust6oBPHMyMOGeNFUXYBwoQs2M1tukTme6E5apUYGlr6LPpCClU/IDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T07:26:01.484571Z"},"content_sha256":"a05a0890dab385c105c276393066ee902a05ea66cdb992430cd7cc0f57a98ca3","schema_version":"1.0","event_id":"sha256:a05a0890dab385c105c276393066ee902a05ea66cdb992430cd7cc0f57a98ca3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4MCGDEXWGX3XVO4E6X5JCBEQOB/bundle.json","state_url":"https://pith.science/pith/4MCGDEXWGX3XVO4E6X5JCBEQOB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4MCGDEXWGX3XVO4E6X5JCBEQOB/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":"2026-06-30T07:26:01Z","links":{"resolver":"https://pith.science/pith/4MCGDEXWGX3XVO4E6X5JCBEQOB","bundle":"https://pith.science/pith/4MCGDEXWGX3XVO4E6X5JCBEQOB/bundle.json","state":"https://pith.science/pith/4MCGDEXWGX3XVO4E6X5JCBEQOB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4MCGDEXWGX3XVO4E6X5JCBEQOB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:4MCGDEXWGX3XVO4E6X5JCBEQOB","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":"98608f1192791812afbae8ebfafa7a9d3ea5103e7a9b049af11ab8bdcc53c727","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-11-07T02:23:46Z","title_canon_sha256":"6bc908f07e0469d6f533d4cdc015d66eb83fc755a3e4afd2cea39fbbe93d0012"},"schema_version":"1.0","source":{"id":"1711.02261","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1711.02261","created_at":"2026-05-18T00:31:08Z"},{"alias_kind":"arxiv_version","alias_value":"1711.02261v1","created_at":"2026-05-18T00:31:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.02261","created_at":"2026-05-18T00:31:08Z"},{"alias_kind":"pith_short_12","alias_value":"4MCGDEXWGX3X","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_16","alias_value":"4MCGDEXWGX3XVO4E","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_8","alias_value":"4MCGDEXW","created_at":"2026-05-18T12:31:00Z"}],"graph_snapshots":[{"event_id":"sha256:a05a0890dab385c105c276393066ee902a05ea66cdb992430cd7cc0f57a98ca3","target":"graph","created_at":"2026-05-18T00:31:08Z","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":"Ancient solutions arise in the study of parabolic blow-ups. If we can categorize ancient solutions, we can better understand blow-up limits. Based on an argument of Giga and Kohn, we give a Liouville-type theorem restricting ancient, type-I, non-collapsing two- dimensional mean curvature flows to either spheres or cylinders.","authors_text":"Kevin Sonnanburg","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-11-07T02:23:46Z","title":"A Liouville Theorem for Mean Curvature Flow"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.02261","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:bb7cc2906271b2027a1a8d80d077fee6cbdb402c71e8543de4eb681c299c0d24","target":"record","created_at":"2026-05-18T00:31:08Z","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":"98608f1192791812afbae8ebfafa7a9d3ea5103e7a9b049af11ab8bdcc53c727","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-11-07T02:23:46Z","title_canon_sha256":"6bc908f07e0469d6f533d4cdc015d66eb83fc755a3e4afd2cea39fbbe93d0012"},"schema_version":"1.0","source":{"id":"1711.02261","kind":"arxiv","version":1}},"canonical_sha256":"e3046192f635f77abb84f5fa910490706b82ec582586606f7c1f9682280c2b1a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e3046192f635f77abb84f5fa910490706b82ec582586606f7c1f9682280c2b1a","first_computed_at":"2026-05-18T00:31:08.397163Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:31:08.397163Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IkywI4Xqhh7Re2pc/JPNEIXHpbexX1AUx0PcR/8Xi1SKql0EPd9JsuZSvUuxmQF/6VhpQC46hlfcPL4ApiJ7Bg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:31:08.397844Z","signed_message":"canonical_sha256_bytes"},"source_id":"1711.02261","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bb7cc2906271b2027a1a8d80d077fee6cbdb402c71e8543de4eb681c299c0d24","sha256:a05a0890dab385c105c276393066ee902a05ea66cdb992430cd7cc0f57a98ca3"],"state_sha256":"b959d3c32279f59298ad6bb0095294aabec9314570f204eb78f6d8a83fc00ad1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HJAfrJvc+WuCRFGJ4ifzRdxoMPULm79hMOJHK5nxiJC460gctO0WDYLronQLPRER3hBDKmKHJ/bp3OxD04QyBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-30T07:26:01.486575Z","bundle_sha256":"8250ba7362e048aadcedf79b7917aab877edcee4d002d0b4db9008a399310730"}}