{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:RTPDNLLB7FN375WSBLF2KPLA7U","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":"9fd5cc94de5f6e7317b6303790322acc3e47a80b2ac443396e31f50e6e168623","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-09-25T15:15:33Z","title_canon_sha256":"b48bc440d230b13518714336f5cb768d74c6aeee265ea86452cbab5c1e457bf1"},"schema_version":"1.0","source":{"id":"1209.5640","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.5640","created_at":"2026-05-18T00:14:16Z"},{"alias_kind":"arxiv_version","alias_value":"1209.5640v2","created_at":"2026-05-18T00:14:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.5640","created_at":"2026-05-18T00:14:16Z"},{"alias_kind":"pith_short_12","alias_value":"RTPDNLLB7FN3","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_16","alias_value":"RTPDNLLB7FN375WS","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_8","alias_value":"RTPDNLLB","created_at":"2026-05-18T12:27:20Z"}],"graph_snapshots":[{"event_id":"sha256:020a4bd25d28a661b2568c3dd67427689ad88f9f5d103d011f891f11b2226621","target":"graph","created_at":"2026-05-18T00:14:16Z","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 prove that, for general cost functions on $\\mathbb{R}^n$, or for the cost $d^2/2$ on a Riemannian manifold, optimal transport maps between smooth densities are always smooth outside a closed singular set of measure zero.","authors_text":"Alessio Figalli, Guido De Philippis","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-09-25T15:15:33Z","title":"Partial Regularity for optimal transport maps"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.5640","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":"sha256:121cb9d6a7cb0e03f441a65e960db9e564ca85931d74556203e463a79f7da602","target":"record","created_at":"2026-05-18T00:14:16Z","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":"9fd5cc94de5f6e7317b6303790322acc3e47a80b2ac443396e31f50e6e168623","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-09-25T15:15:33Z","title_canon_sha256":"b48bc440d230b13518714336f5cb768d74c6aeee265ea86452cbab5c1e457bf1"},"schema_version":"1.0","source":{"id":"1209.5640","kind":"arxiv","version":2}},"canonical_sha256":"8cde36ad61f95bbff6d20acba53d60fd22e9faeb47f42b9e97728cdffdb9a436","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8cde36ad61f95bbff6d20acba53d60fd22e9faeb47f42b9e97728cdffdb9a436","first_computed_at":"2026-05-18T00:14:16.882471Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:14:16.882471Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vntnrn/AhHTRwy5zJRN/g9LJM+gFnUkeSX9p/uumj2N3IzGCCsht2jqD67FlN0IfFGtc7Mde4IekeezDSq6NAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:14:16.883054Z","signed_message":"canonical_sha256_bytes"},"source_id":"1209.5640","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:121cb9d6a7cb0e03f441a65e960db9e564ca85931d74556203e463a79f7da602","sha256:020a4bd25d28a661b2568c3dd67427689ad88f9f5d103d011f891f11b2226621"],"state_sha256":"a7b2e9163216c30f80640b81a07892df46718716d893a198c826b943c4604447"}