{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:75ECLKHW3F4I3G4L7TLPAID3N7","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":"dbaed60ed44eed3e64f6967c13d2447444a703754c75902394607aeb6d012272","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2025-06-18T10:12:49Z","title_canon_sha256":"044c96af71305c6cfe5da230bc2e61f230d91137220854f817650d976df48e8c"},"schema_version":"1.0","source":{"id":"2506.15333","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.15333","created_at":"2026-07-05T11:23:41Z"},{"alias_kind":"arxiv_version","alias_value":"2506.15333v1","created_at":"2026-07-05T11:23:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.15333","created_at":"2026-07-05T11:23:41Z"},{"alias_kind":"pith_short_12","alias_value":"75ECLKHW3F4I","created_at":"2026-07-05T11:23:41Z"},{"alias_kind":"pith_short_16","alias_value":"75ECLKHW3F4I3G4L","created_at":"2026-07-05T11:23:41Z"},{"alias_kind":"pith_short_8","alias_value":"75ECLKHW","created_at":"2026-07-05T11:23:41Z"}],"graph_snapshots":[{"event_id":"sha256:cab4e59c60b2f4b86ee7672921004c9a00e7ea2c7fc636a6f8eba04b76d5324f","target":"graph","created_at":"2026-07-05T11:23:41Z","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/2506.15333/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Representation results for absolutely continuous curves $\\mu:[0,T]\\to \\mathcal{P}_p(\\mathbb{R}^d)$, $p>1$, with values in the Wasserstein space $(\\mathcal{P}_p(\\mathbb{R}^d),W_p)$ of Borel probability measures in $\\mathbb{R}^d$ with finite $p$-moment, provide a crucial tool to study evolutionary PDEs in a measure-theoretic setting. They are strictly related to the superposition principle for measure-valued solutions to the continuity equation. This paper addresses the extension of these results to the case $p=1$, and to curves $\\mu:[0,+\\infty)\\to\\mathcal{P}_1(\\mathbb{R}^d)$ that are only of bo","authors_text":"Giuseppe Savar\\'e, Riccarda Rossi, Stefano Almi","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2025-06-18T10:12:49Z","title":"The superposition principle for the continuity equation with singular flux"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.15333","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":"sha256:177884e566fdfa69ae44b9f54c9f9156c0113b21c6bd55caf443731acfa5bcec","target":"record","created_at":"2026-07-05T11:23:41Z","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":"dbaed60ed44eed3e64f6967c13d2447444a703754c75902394607aeb6d012272","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2025-06-18T10:12:49Z","title_canon_sha256":"044c96af71305c6cfe5da230bc2e61f230d91137220854f817650d976df48e8c"},"schema_version":"1.0","source":{"id":"2506.15333","kind":"arxiv","version":1}},"canonical_sha256":"ff4825a8f6d9788d9b8bfcd6f0207b6ffdbb76bc3b91927df10a84c0a195d6f4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ff4825a8f6d9788d9b8bfcd6f0207b6ffdbb76bc3b91927df10a84c0a195d6f4","first_computed_at":"2026-07-05T11:23:41.514500Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:23:41.514500Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"x4BS9vblx6xCSYnbEW2Vq5frL8Kii4zHy959DJMlvw93SsghkCBXywg++gvcYW4W2+JDq3G/bmJwJSswRtcNAw==","signature_status":"signed_v1","signed_at":"2026-07-05T11:23:41.515192Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.15333","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:177884e566fdfa69ae44b9f54c9f9156c0113b21c6bd55caf443731acfa5bcec","sha256:cab4e59c60b2f4b86ee7672921004c9a00e7ea2c7fc636a6f8eba04b76d5324f"],"state_sha256":"192edfeea151fdb7d06abf2767b6d5299de96aa515ae180154dc22609b4ab938"}