{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:WHKQ4IXW6GWL4YHYULGSM6E3BD","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":"da128f8f41f5367be2a40d553c164e740cc936296813b69222db19f3e154b89d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2015-05-03T11:55:01Z","title_canon_sha256":"3c1ca804f4bfdb49c97e6b08b1ea15403c65a7a89e17b1b00e4e44f635f45d9e"},"schema_version":"1.0","source":{"id":"1505.00420","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.00420","created_at":"2026-05-18T00:10:14Z"},{"alias_kind":"arxiv_version","alias_value":"1505.00420v3","created_at":"2026-05-18T00:10:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.00420","created_at":"2026-05-18T00:10:14Z"},{"alias_kind":"pith_short_12","alias_value":"WHKQ4IXW6GWL","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"WHKQ4IXW6GWL4YHY","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"WHKQ4IXW","created_at":"2026-05-18T12:29:47Z"}],"graph_snapshots":[{"event_id":"sha256:15c6ab4d8e7bdd358f27d30603b6a934393784bd77f49d0ba3c93ba0a34e69df","target":"graph","created_at":"2026-05-18T00:10:14Z","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":"In this paper, we give the characterization of metric measure spaces that satisfy synthetic lower Riemannian Ricci curvature bounds (so called $RCD^*(K,N)$ spaces) with \\emph{non-empty} one dimensional regular sets. In particular, we prove that the class of Ricci limit spaces with $Ric \\ge K$ and Hausdorff dimension $N$ and the class of $RCD^*(K,N)$ spaces coincide for $N < 2$ (They can be either complete intervals or circles). We will also prove a Bishop-Gromov type inequality ( that is ,roughly speaking, a converse to the L\\'{e}vy-Gromov's isoperimetric inequality and was previously only kno","authors_text":"Sajjad Lakzian, Yu Kitabeppu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2015-05-03T11:55:01Z","title":"Characterization of Low Dimensional $RCD^*(K,N)$ spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.00420","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":"sha256:13c5abe7a276a2846e4f375065367dc213fdb5d09c83fc8dab35d277af766d55","target":"record","created_at":"2026-05-18T00:10:14Z","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":"da128f8f41f5367be2a40d553c164e740cc936296813b69222db19f3e154b89d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2015-05-03T11:55:01Z","title_canon_sha256":"3c1ca804f4bfdb49c97e6b08b1ea15403c65a7a89e17b1b00e4e44f635f45d9e"},"schema_version":"1.0","source":{"id":"1505.00420","kind":"arxiv","version":3}},"canonical_sha256":"b1d50e22f6f1acbe60f8a2cd26789b08f46bd9f8c805922e4166db4dbf92d45d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b1d50e22f6f1acbe60f8a2cd26789b08f46bd9f8c805922e4166db4dbf92d45d","first_computed_at":"2026-05-18T00:10:14.784399Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:10:14.784399Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"J4TuZJVOl/918MeD4LSMUt8jWCN9Cna+gJiCdA3Krn87zMaZsyZrGE/RGY5vhO3wsRDGrPNEjEFSulEWHeLIDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:10:14.784984Z","signed_message":"canonical_sha256_bytes"},"source_id":"1505.00420","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:13c5abe7a276a2846e4f375065367dc213fdb5d09c83fc8dab35d277af766d55","sha256:15c6ab4d8e7bdd358f27d30603b6a934393784bd77f49d0ba3c93ba0a34e69df"],"state_sha256":"b9acb1f87002bbf684501f9269ab1f14d62d3344d454e3a3b38230feaba4d8f6"}