{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:PXH4I67LQPS4LLMCO23VQPNRCL","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":"fe4af1607f41f9beee9d8ce98168a01c92f9902f8ffb86a071b27902410d1664","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-08-02T02:51:36Z","title_canon_sha256":"379bc3dbf585764369c0f51f5b3dae6a8d098c3d67a2878d3fc06b91b96e1ab6"},"schema_version":"1.0","source":{"id":"2608.00944","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.00944","created_at":"2026-08-04T01:55:53Z"},{"alias_kind":"arxiv_version","alias_value":"2608.00944v1","created_at":"2026-08-04T01:55:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.00944","created_at":"2026-08-04T01:55:53Z"},{"alias_kind":"pith_short_12","alias_value":"PXH4I67LQPS4","created_at":"2026-08-04T01:55:53Z"},{"alias_kind":"pith_short_16","alias_value":"PXH4I67LQPS4LLMC","created_at":"2026-08-04T01:55:53Z"},{"alias_kind":"pith_short_8","alias_value":"PXH4I67L","created_at":"2026-08-04T01:55:53Z"}],"graph_snapshots":[{"event_id":"sha256:1fca430af6fd5e4f0c32a4f0b71c7b739d995e124e03c6c79d3b767abec5609d","target":"graph","created_at":"2026-08-04T01:55:53Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2608.00944/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let \\((M^m,g)\\) be a closed Riemannian manifold with \\(\\sec_g\\leq-1\\). We prove that the bottom spectrum of its universal cover attains McKean's lower bound if and only if the universal cover is hyperbolic space of constant sectional curvature \\(-1\\). More generally, for every \\(1<p<\\infty\\), the variational \\(p\\)-fundamental tone satisfies \\[\n  \\lambda_{1,p}(\\wti M)\n  \\geq\\left(\\frac{m-1}{p}\\right)^p, \\] and equality for some \\(p\\in(1,\\infty)\\) holds if and only if \\(\\wti M\\cong\\bH^m(-1)\\). In that case, equality holds for every \\(p\\in(1,\\infty)\\). The proof converts the two McKean defects of","authors_text":"Bo Zhu, Kuntao Jin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-08-02T02:51:36Z","title":"McKean Rigidity for Cocompact Negatively Curved Manifolds and the \\(p\\)-Laplacian"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.00944","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":"sha256:78a410f6eb4d4f0ff46d75ef2d0b70291859495e3d1988768f637a457be35b97","target":"record","created_at":"2026-08-04T01:55:53Z","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":"fe4af1607f41f9beee9d8ce98168a01c92f9902f8ffb86a071b27902410d1664","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-08-02T02:51:36Z","title_canon_sha256":"379bc3dbf585764369c0f51f5b3dae6a8d098c3d67a2878d3fc06b91b96e1ab6"},"schema_version":"1.0","source":{"id":"2608.00944","kind":"arxiv","version":1}},"canonical_sha256":"7dcfc47beb83e5c5ad8276b7583db112d1cb02e21457a7c638900afbaec77720","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7dcfc47beb83e5c5ad8276b7583db112d1cb02e21457a7c638900afbaec77720","first_computed_at":"2026-08-04T01:55:53.761927Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-08-04T01:55:53.761927Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"a4aiAYpkDptReZuaCAfMeZm4RrBgSzQmeM8MX37b/C4WqiZIvOduyjuZ7eRz6R4pYmmKsjZSSuhnXR9LHMQ7Aw==","signature_status":"signed_v1","signed_at":"2026-08-04T01:55:53.763413Z","signed_message":"canonical_sha256_bytes"},"source_id":"2608.00944","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:78a410f6eb4d4f0ff46d75ef2d0b70291859495e3d1988768f637a457be35b97","sha256:1fca430af6fd5e4f0c32a4f0b71c7b739d995e124e03c6c79d3b767abec5609d"],"state_sha256":"b0967cb5d05f19215db0011c778ecc013d0ec3a2c61e1f13b19fa0b035ec0d7d"}