{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:MDAJSRO5DNNGEYWMCKAIJCMEXZ","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":"89df9bc6be4792db7b294c3710fa2591abc8a09b1e4450df441f358d9c7b2526","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-06-01T19:47:08Z","title_canon_sha256":"0f5d885acc7071067e73285cff7c845dae1d7102526c6c2c3bca380547913e3a"},"schema_version":"1.0","source":{"id":"2606.02829","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.02829","created_at":"2026-06-03T01:05:24Z"},{"alias_kind":"arxiv_version","alias_value":"2606.02829v1","created_at":"2026-06-03T01:05:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.02829","created_at":"2026-06-03T01:05:24Z"},{"alias_kind":"pith_short_12","alias_value":"MDAJSRO5DNNG","created_at":"2026-06-03T01:05:24Z"},{"alias_kind":"pith_short_16","alias_value":"MDAJSRO5DNNGEYWM","created_at":"2026-06-03T01:05:24Z"},{"alias_kind":"pith_short_8","alias_value":"MDAJSRO5","created_at":"2026-06-03T01:05:24Z"}],"graph_snapshots":[{"event_id":"sha256:3608a949e342027ac38b9faae65eff65694e49057c1f07f0fbf38bdba4bbc201","target":"graph","created_at":"2026-06-03T01:05:24Z","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/2606.02829/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove a sharp Clifford-threshold focal-radius estimate and rigidity for immersed hypersurfaces. Under a $p$-form curvature condition, formulated by the Weitzenb\\\"ock curvature term together with $\\mathrm{Ric}_p\\ge p$, any closed two-sided immersion $F:\\Sigma^m\\to M^{m+1}$ with $b_p(\\Sigma;\\mathbb R)\\neq0$ and $1\\le p\\le m/2$ satisfies \\[\n  r_f(F,M)\\le\\frac{\\pi}{4}. \\] The equality case is rigid: if the ambient manifold is complete, equality forces the hypersurface to be locally the Clifford hypersurface $S^p(1/\\sqrt2)\\times S^{m-p}(1/\\sqrt2)\\subset S^{m+1}(1)$; if the ambient manifold is co","authors_text":"Jingbo Wan, Tsz-Kiu Aaron Chow","cross_cats":["math.GT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-06-01T19:47:08Z","title":"Sharp focal radius estimate and rigidity of hypersurfaces in manifolds with positive curvature"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.02829","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":"sha256:235d0ceac51102dd2eb44d70be2e1c1b68a8a65a9190936abb5602dd2521b288","target":"record","created_at":"2026-06-03T01:05:24Z","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":"89df9bc6be4792db7b294c3710fa2591abc8a09b1e4450df441f358d9c7b2526","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-06-01T19:47:08Z","title_canon_sha256":"0f5d885acc7071067e73285cff7c845dae1d7102526c6c2c3bca380547913e3a"},"schema_version":"1.0","source":{"id":"2606.02829","kind":"arxiv","version":1}},"canonical_sha256":"60c09945dd1b5a6262cc1280848984be7355b1b048baa27134450f970c9b599e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"60c09945dd1b5a6262cc1280848984be7355b1b048baa27134450f970c9b599e","first_computed_at":"2026-06-03T01:05:24.055880Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-03T01:05:24.055880Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qsNXmdS0vn3ZJje/Ta1vtAbKD/qgEa2f76XVO1Urt5c4SIrqOGxPRV96IN9BH3EnAQTXDojN1fGsNZr14L1EDg==","signature_status":"signed_v1","signed_at":"2026-06-03T01:05:24.056281Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.02829","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:235d0ceac51102dd2eb44d70be2e1c1b68a8a65a9190936abb5602dd2521b288","sha256:3608a949e342027ac38b9faae65eff65694e49057c1f07f0fbf38bdba4bbc201"],"state_sha256":"b66b6e1444c498098e87c326ab9cda9229ab247bac15e205bb3fc87bd3ee06cf"}