{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:RAYQGZVR3G44YUK2YO3U5SMYXF","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":"706f74651e917ada6a7423b5c26d657ff0fedcab8dfe3bb21185213a85b6d649","cross_cats_sorted":["math.DG","math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2019-02-03T18:07:49Z","title_canon_sha256":"625efcacb110c1ebc15c46e054ab7b2a96dc56388b099196cbde1d987018ce28"},"schema_version":"1.0","source":{"id":"1902.00942","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1902.00942","created_at":"2026-07-05T03:05:40Z"},{"alias_kind":"arxiv_version","alias_value":"1902.00942v3","created_at":"2026-07-05T03:05:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1902.00942","created_at":"2026-07-05T03:05:40Z"},{"alias_kind":"pith_short_12","alias_value":"RAYQGZVR3G44","created_at":"2026-07-05T03:05:40Z"},{"alias_kind":"pith_short_16","alias_value":"RAYQGZVR3G44YUK2","created_at":"2026-07-05T03:05:40Z"},{"alias_kind":"pith_short_8","alias_value":"RAYQGZVR","created_at":"2026-07-05T03:05:40Z"}],"graph_snapshots":[{"event_id":"sha256:2a90c29f615b79889f2e3aa3a6a116d6ae984281cbfc7deb28bd58c4856db106","target":"graph","created_at":"2026-07-05T03:05:40Z","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/1902.00942/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper we investigate Lott-Sturm-Villani's synthetic lower Ricci curvature bound on Riemannian manifolds with boundary. We prove several measure rigidity results for some important functional and geometric inequalities, which completely characterize ${\\rm CD}(K, \\infty)$ condition and non-collapsed ${\\rm CD}(K, N)$ condition on Riemannian manifolds with boundary. In particular, using $L^1$-optimal transportation theory, we prove that ${\\rm CD}(K, \\infty)$ condition implies geodesical convexity.","authors_text":"Bang-Xian Han","cross_cats":["math.DG","math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2019-02-03T18:07:49Z","title":"Measure rigidity of synthetic lower Ricci curvature bound on Riemannian manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.00942","kind":"arxiv","version":3},"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":"sha256:2abca37c84416c9c65d61f74c3e62818e693d90dabd7619d26f8fa8814e326d1","target":"record","created_at":"2026-07-05T03:05:40Z","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":"706f74651e917ada6a7423b5c26d657ff0fedcab8dfe3bb21185213a85b6d649","cross_cats_sorted":["math.DG","math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2019-02-03T18:07:49Z","title_canon_sha256":"625efcacb110c1ebc15c46e054ab7b2a96dc56388b099196cbde1d987018ce28"},"schema_version":"1.0","source":{"id":"1902.00942","kind":"arxiv","version":3}},"canonical_sha256":"88310366b1d9b9cc515ac3b74ec998b95e068666ef1a862b3f93dfc1fabc0af6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"88310366b1d9b9cc515ac3b74ec998b95e068666ef1a862b3f93dfc1fabc0af6","first_computed_at":"2026-07-05T03:05:40.971068Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:05:40.971068Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AGNyGttJS9JmJuZW2ojtI/Rwhue33AsHyvDKGV+m0L3T/RdQEYCjhDf6+LaVspsbweNkTn5dh+EiXrCpkBUbCg==","signature_status":"signed_v1","signed_at":"2026-07-05T03:05:40.971535Z","signed_message":"canonical_sha256_bytes"},"source_id":"1902.00942","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2abca37c84416c9c65d61f74c3e62818e693d90dabd7619d26f8fa8814e326d1","sha256:2a90c29f615b79889f2e3aa3a6a116d6ae984281cbfc7deb28bd58c4856db106"],"state_sha256":"5aa2ceb39b3064a472c7d7e5f167d6c22014915d5f4673a9378dbb71f8f06076"}