{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:W6OUASKVKCY73222Y43GDCJU23","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":"ea1a69a5864017bf46fbe9071fde54d64c9f21ebc4d49e9395870b3faf90d764","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-08-23T23:17:49Z","title_canon_sha256":"dd36b01d96eb454fc0375b4d47a4bb956e2abc1977f382f7697619f44232441f"},"schema_version":"1.0","source":{"id":"1408.5534","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1408.5534","created_at":"2026-05-18T02:32:53Z"},{"alias_kind":"arxiv_version","alias_value":"1408.5534v2","created_at":"2026-05-18T02:32:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.5534","created_at":"2026-05-18T02:32:53Z"},{"alias_kind":"pith_short_12","alias_value":"W6OUASKVKCY7","created_at":"2026-05-18T12:28:54Z"},{"alias_kind":"pith_short_16","alias_value":"W6OUASKVKCY73222","created_at":"2026-05-18T12:28:54Z"},{"alias_kind":"pith_short_8","alias_value":"W6OUASKV","created_at":"2026-05-18T12:28:54Z"}],"graph_snapshots":[{"event_id":"sha256:01512236ffa11a87b05ddf2e9827993859321d7396c6a513a9e5c9de8c48a599","target":"graph","created_at":"2026-05-18T02:32:53Z","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 give a characterization of critical points that allows us to define a metric invariant on all Riemannian manifolds $M$ with a lower sectional curvature bound and an upper radius bound. We show there is a uniform upper volume bound for all such manifolds with an upper bound on this invariant. We generalize results by Grove and Petersen and by Sill, Wilhelm, and the author by showing any such $M$ that has volume sufficiently close to this upper bound is diffeomorphic to the standard sphere $S^{n}$ or a standard lens space $S^n/\\mathbb{Z}_m$ where $m\\in\\{2,3,\\ldots\\}$ is no larger than an a pr","authors_text":"Curtis Pro","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-08-23T23:17:49Z","title":"Sagitta, Lenses, and Maximal Volume"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.5534","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":"sha256:958976e7b9cbed7cbed488d2c80b8f054e229d7ef6b98614ab93df4987375107","target":"record","created_at":"2026-05-18T02:32:53Z","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":"ea1a69a5864017bf46fbe9071fde54d64c9f21ebc4d49e9395870b3faf90d764","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-08-23T23:17:49Z","title_canon_sha256":"dd36b01d96eb454fc0375b4d47a4bb956e2abc1977f382f7697619f44232441f"},"schema_version":"1.0","source":{"id":"1408.5534","kind":"arxiv","version":2}},"canonical_sha256":"b79d40495550b1fdeb5ac736618934d6d13f18160079ad7dbb5e5b29ad4623f8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b79d40495550b1fdeb5ac736618934d6d13f18160079ad7dbb5e5b29ad4623f8","first_computed_at":"2026-05-18T02:32:53.370289Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:32:53.370289Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Mc+9/WHOoHZo0PZ/NANECWs9CwosDexs11OZrTnoE9jrA4yFJFJwHKcX0FA1/Pl69BRSiNOImesqDiKp6xIWCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:32:53.370967Z","signed_message":"canonical_sha256_bytes"},"source_id":"1408.5534","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:958976e7b9cbed7cbed488d2c80b8f054e229d7ef6b98614ab93df4987375107","sha256:01512236ffa11a87b05ddf2e9827993859321d7396c6a513a9e5c9de8c48a599"],"state_sha256":"753b758655d406bfd8da9148c5e5def95bf2d748733f07fc4405f0d32230fde0"}