{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:OTARS4D26WNKO3E6UKNK2AHYXE","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":"a601b9c73a51a14ad35345b9154fbc550f0f184ab66649f0408bb8d2be68f05a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-06-16T12:20:42Z","title_canon_sha256":"8758321aecf297660d25eefc8229c8de3ace66e139c00e6d2d3188732c59bc44"},"schema_version":"1.0","source":{"id":"1506.04936","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.04936","created_at":"2026-05-17T23:55:43Z"},{"alias_kind":"arxiv_version","alias_value":"1506.04936v2","created_at":"2026-05-17T23:55:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.04936","created_at":"2026-05-17T23:55:43Z"},{"alias_kind":"pith_short_12","alias_value":"OTARS4D26WNK","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_16","alias_value":"OTARS4D26WNKO3E6","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_8","alias_value":"OTARS4D2","created_at":"2026-05-18T12:29:34Z"}],"graph_snapshots":[{"event_id":"sha256:0727234aa45b1933235656836f5ccaf8910d7929d3cd67bb69b956b087527d1b","target":"graph","created_at":"2026-05-17T23:55:43Z","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 a compact $RCD^*(0,N)$ (or equivalently $RCD(0,N)$) metric measure space, $\\left(X, d, m \\right)$, with $\\diam X \\le d$ and its first (nonzero) eigenvalue of the Laplacian (in the sense of Ambrosio-Gigli-Savar\\'{e}) , $\\lambda_1 = \\frac{\\pi^2}{d^2}$, has to be a circle or a line segment with diameter, $\\pi$. This completely characterizes the equality in Zhong-Yang type sharp spectral gap estimates in the metric measure setting with Riemannian lower Ricci bounds. Among such spaces, are the familiar Riemannian manifolds with $\\Ric \\ge 0$, $(0,N)-$ Bakry-\\'{E}mery manifolds, $(0,n)-","authors_text":"Sajjad Lakzian","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-06-16T12:20:42Z","title":"Characterization of equality in Zhong-Yang type (sharp) spectral gap estimates for metric measure spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.04936","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":"sha256:70682dbb3e5d6f7123f517c87a18778d093bed454570347b3cf65df3a79e7732","target":"record","created_at":"2026-05-17T23:55:43Z","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":"a601b9c73a51a14ad35345b9154fbc550f0f184ab66649f0408bb8d2be68f05a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-06-16T12:20:42Z","title_canon_sha256":"8758321aecf297660d25eefc8229c8de3ace66e139c00e6d2d3188732c59bc44"},"schema_version":"1.0","source":{"id":"1506.04936","kind":"arxiv","version":2}},"canonical_sha256":"74c119707af59aa76c9ea29aad00f8b93e44d9026e4d87ab283e2f725f0cf457","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"74c119707af59aa76c9ea29aad00f8b93e44d9026e4d87ab283e2f725f0cf457","first_computed_at":"2026-05-17T23:55:43.504039Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:55:43.504039Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IO9G2rMhaIdSgHJDfRSEJSqq6RIAZY8Al1q9yuvbV/kKYjYERIC0bB6UwzYUYkyjMUw/4QydEn9Q/M/kJoAiAw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:55:43.504583Z","signed_message":"canonical_sha256_bytes"},"source_id":"1506.04936","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:70682dbb3e5d6f7123f517c87a18778d093bed454570347b3cf65df3a79e7732","sha256:0727234aa45b1933235656836f5ccaf8910d7929d3cd67bb69b956b087527d1b"],"state_sha256":"685d2d5d49f403b31b5b2fb0e937e02f2d2c508bc2b4696ee2cff7f751ca19c4"}