{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:6WE4QZEMLYXVO5HC64XYL7TRJV","short_pith_number":"pith:6WE4QZEM","canonical_record":{"source":{"id":"2504.11556","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2025-04-15T18:55:40Z","cross_cats_sorted":[],"title_canon_sha256":"5d07b1cf915e7e10335602be6fb128276aa4d37c7c507fc40c2cf1f66c2c4142","abstract_canon_sha256":"f7cad89f799832164b6d1cd4b90c1c036a8c9bad2c2e1d09279696c95ccb6091"},"schema_version":"1.0"},"canonical_sha256":"f589c8648c5e2f5774e2f72f85fe714d46c6e894cf0b254f64a95cf80a79e944","source":{"kind":"arxiv","id":"2504.11556","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2504.11556","created_at":"2026-07-05T10:49:36Z"},{"alias_kind":"arxiv_version","alias_value":"2504.11556v1","created_at":"2026-07-05T10:49:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2504.11556","created_at":"2026-07-05T10:49:36Z"},{"alias_kind":"pith_short_12","alias_value":"6WE4QZEMLYXV","created_at":"2026-07-05T10:49:36Z"},{"alias_kind":"pith_short_16","alias_value":"6WE4QZEMLYXVO5HC","created_at":"2026-07-05T10:49:36Z"},{"alias_kind":"pith_short_8","alias_value":"6WE4QZEM","created_at":"2026-07-05T10:49:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:6WE4QZEMLYXVO5HC64XYL7TRJV","target":"record","payload":{"canonical_record":{"source":{"id":"2504.11556","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2025-04-15T18:55:40Z","cross_cats_sorted":[],"title_canon_sha256":"5d07b1cf915e7e10335602be6fb128276aa4d37c7c507fc40c2cf1f66c2c4142","abstract_canon_sha256":"f7cad89f799832164b6d1cd4b90c1c036a8c9bad2c2e1d09279696c95ccb6091"},"schema_version":"1.0"},"canonical_sha256":"f589c8648c5e2f5774e2f72f85fe714d46c6e894cf0b254f64a95cf80a79e944","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:49:36.855923Z","signature_b64":"bPHNFfgxu2ANyhLBuq1RsK8lGXz6OoXYULUfM2AP9FpWSrI5DXBev2+4avoZKZuuem6ZoyGj++Jk4BiZLjDQDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f589c8648c5e2f5774e2f72f85fe714d46c6e894cf0b254f64a95cf80a79e944","last_reissued_at":"2026-07-05T10:49:36.855522Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:49:36.855522Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2504.11556","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":"2026-07-05T10:49:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Rel1wMPuHLZ4DZ9Pj4aEQEnznQTAulQeS3musRLTJz7tkCDElIx3QmyA2jn55SrSj0U1aiqo/OEBWfVWgdDVDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T18:22:27.386385Z"},"content_sha256":"9ef4e653ccc6f2bcabbd03dcfeba8d3e867076b3563c1232edecaca9581176d3","schema_version":"1.0","event_id":"sha256:9ef4e653ccc6f2bcabbd03dcfeba8d3e867076b3563c1232edecaca9581176d3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:6WE4QZEMLYXVO5HC64XYL7TRJV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"$C_{loc}^{1,1}$ optimal pairs in the dual optimal transport problem for a Lorentzian cost along displacement interpolations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Alec Metsch","submitted_at":"2025-04-15T18:55:40Z","abstract_excerpt":"We consider the optimal transportation problem on a globally hyperbolic spacetime with a cost function $c$, which corresponds to the optimal transportation problem on a complete Riemannian manifold where the cost function is given by the squared Riemannian distance. Building upon methods of weak KAM theory, we will establish the existence of $C_{loc}^{1,1}$ optimal pairs for the dual optimal transport problem for probability measures along displacement interpolations."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.11556","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/2504.11556/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":"2026-07-05T10:49:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Bw+iCDcV+iPWs+qS6t4mbPoCGzdcWtEmxjxe9AoMvV8/rD2sK89OputkbiOXwgrP/Qxjbf2SvX/yO80BnAyrAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T18:22:27.387152Z"},"content_sha256":"c9df28ed75735372938fa1ffcaade408a7d006a561412d9ec27c82fb18bcac4f","schema_version":"1.0","event_id":"sha256:c9df28ed75735372938fa1ffcaade408a7d006a561412d9ec27c82fb18bcac4f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6WE4QZEMLYXVO5HC64XYL7TRJV/bundle.json","state_url":"https://pith.science/pith/6WE4QZEMLYXVO5HC64XYL7TRJV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6WE4QZEMLYXVO5HC64XYL7TRJV/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":"2026-08-14T18:22:27Z","links":{"resolver":"https://pith.science/pith/6WE4QZEMLYXVO5HC64XYL7TRJV","bundle":"https://pith.science/pith/6WE4QZEMLYXVO5HC64XYL7TRJV/bundle.json","state":"https://pith.science/pith/6WE4QZEMLYXVO5HC64XYL7TRJV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6WE4QZEMLYXVO5HC64XYL7TRJV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:6WE4QZEMLYXVO5HC64XYL7TRJV","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":"f7cad89f799832164b6d1cd4b90c1c036a8c9bad2c2e1d09279696c95ccb6091","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2025-04-15T18:55:40Z","title_canon_sha256":"5d07b1cf915e7e10335602be6fb128276aa4d37c7c507fc40c2cf1f66c2c4142"},"schema_version":"1.0","source":{"id":"2504.11556","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2504.11556","created_at":"2026-07-05T10:49:36Z"},{"alias_kind":"arxiv_version","alias_value":"2504.11556v1","created_at":"2026-07-05T10:49:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2504.11556","created_at":"2026-07-05T10:49:36Z"},{"alias_kind":"pith_short_12","alias_value":"6WE4QZEMLYXV","created_at":"2026-07-05T10:49:36Z"},{"alias_kind":"pith_short_16","alias_value":"6WE4QZEMLYXVO5HC","created_at":"2026-07-05T10:49:36Z"},{"alias_kind":"pith_short_8","alias_value":"6WE4QZEM","created_at":"2026-07-05T10:49:36Z"}],"graph_snapshots":[{"event_id":"sha256:c9df28ed75735372938fa1ffcaade408a7d006a561412d9ec27c82fb18bcac4f","target":"graph","created_at":"2026-07-05T10:49:36Z","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/2504.11556/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We consider the optimal transportation problem on a globally hyperbolic spacetime with a cost function $c$, which corresponds to the optimal transportation problem on a complete Riemannian manifold where the cost function is given by the squared Riemannian distance. Building upon methods of weak KAM theory, we will establish the existence of $C_{loc}^{1,1}$ optimal pairs for the dual optimal transport problem for probability measures along displacement interpolations.","authors_text":"Alec Metsch","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2025-04-15T18:55:40Z","title":"$C_{loc}^{1,1}$ optimal pairs in the dual optimal transport problem for a Lorentzian cost along displacement interpolations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.11556","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":"sha256:9ef4e653ccc6f2bcabbd03dcfeba8d3e867076b3563c1232edecaca9581176d3","target":"record","created_at":"2026-07-05T10:49:36Z","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":"f7cad89f799832164b6d1cd4b90c1c036a8c9bad2c2e1d09279696c95ccb6091","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2025-04-15T18:55:40Z","title_canon_sha256":"5d07b1cf915e7e10335602be6fb128276aa4d37c7c507fc40c2cf1f66c2c4142"},"schema_version":"1.0","source":{"id":"2504.11556","kind":"arxiv","version":1}},"canonical_sha256":"f589c8648c5e2f5774e2f72f85fe714d46c6e894cf0b254f64a95cf80a79e944","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f589c8648c5e2f5774e2f72f85fe714d46c6e894cf0b254f64a95cf80a79e944","first_computed_at":"2026-07-05T10:49:36.855522Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:49:36.855522Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bPHNFfgxu2ANyhLBuq1RsK8lGXz6OoXYULUfM2AP9FpWSrI5DXBev2+4avoZKZuuem6ZoyGj++Jk4BiZLjDQDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T10:49:36.855923Z","signed_message":"canonical_sha256_bytes"},"source_id":"2504.11556","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9ef4e653ccc6f2bcabbd03dcfeba8d3e867076b3563c1232edecaca9581176d3","sha256:c9df28ed75735372938fa1ffcaade408a7d006a561412d9ec27c82fb18bcac4f"],"state_sha256":"790630f468b5622dece202c55e1ff81e8bc6dea809ef32d396f65de7a471c4fc"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+cpWhpvAI2zcGey1IA2Pvu6sAmBX6ylm5dOO581oWwhRh/mOyBZVCjBqYtos/ytU41pQ0Vh0oNGDdHbjQyrECw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-14T18:22:27.394433Z","bundle_sha256":"1d852a7786476919fe610ba99650a4b791e7ee5f3a7d800012049a0e5f6102c7"}}