{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:OK4PLHYSSM3LMWNRV2VJI2RDBF","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":"ab8b0095f65ada009baa67b88013935bf2d49ae257beba4a1896c569f0df6b2b","cross_cats_sorted":["math.AP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2024-12-17T09:20:55Z","title_canon_sha256":"a82247e9021727dabbff04080f23992a3c5d5fec8d9da7bcb1f1afc1ed1cdb19"},"schema_version":"1.0","source":{"id":"2412.12707","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.12707","created_at":"2026-07-05T09:50:19Z"},{"alias_kind":"arxiv_version","alias_value":"2412.12707v1","created_at":"2026-07-05T09:50:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.12707","created_at":"2026-07-05T09:50:19Z"},{"alias_kind":"pith_short_12","alias_value":"OK4PLHYSSM3L","created_at":"2026-07-05T09:50:19Z"},{"alias_kind":"pith_short_16","alias_value":"OK4PLHYSSM3LMWNR","created_at":"2026-07-05T09:50:19Z"},{"alias_kind":"pith_short_8","alias_value":"OK4PLHYS","created_at":"2026-07-05T09:50:19Z"}],"graph_snapshots":[{"event_id":"sha256:1a5aafecc40e161cc375d6c438e8fd788658bd7aee67fa449edab3cea879aa1f","target":"graph","created_at":"2026-07-05T09:50:19Z","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/2412.12707/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove a sharp spectral generalization of the Cheeger--Gromoll splitting theorem. We show that if a complete non-compact Riemannian manifold $M$ of dimension $n\\geq 2$ has at least two ends and\n  \\[\n  \\lambda_1(-\\gamma\\Delta+\\mathrm{Ric})\\geq 0,\n  \\]\n  for some $\\gamma<\\frac{4}{n-1}$, then $M$ splits isometrically as $\\mathbb R\\times N$ for some compact manifold $N$ with nonnegative Ricci curvature. We show that the constant $\\frac{4}{n-1}$ is sharp, and the multiple-end assumption is necessary for any $\\gamma>0$.","authors_text":"Gioacchino Antonelli, Kai Xu, Marco Pozzetta","cross_cats":["math.AP"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2024-12-17T09:20:55Z","title":"A sharp spectral splitting theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.12707","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":"sha256:70976a9a2794ea57e9458f2bf48b7598e86c4b4e94a60b571e72457a095664f0","target":"record","created_at":"2026-07-05T09:50:19Z","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":"ab8b0095f65ada009baa67b88013935bf2d49ae257beba4a1896c569f0df6b2b","cross_cats_sorted":["math.AP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2024-12-17T09:20:55Z","title_canon_sha256":"a82247e9021727dabbff04080f23992a3c5d5fec8d9da7bcb1f1afc1ed1cdb19"},"schema_version":"1.0","source":{"id":"2412.12707","kind":"arxiv","version":1}},"canonical_sha256":"72b8f59f129336b659b1aeaa946a23096ba76101c7bb9afba5cd4583d50b7612","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"72b8f59f129336b659b1aeaa946a23096ba76101c7bb9afba5cd4583d50b7612","first_computed_at":"2026-07-05T09:50:19.559766Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:50:19.559766Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0kKC+c7nIEYpFt4QMXRpdr0/Lrx9c8Bn0T/Fp7ZhT2X2ajgYKN4fjx2TIAsvQes0Mu3GJwLxHpHT5pZjQHjwCw==","signature_status":"signed_v1","signed_at":"2026-07-05T09:50:19.560242Z","signed_message":"canonical_sha256_bytes"},"source_id":"2412.12707","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:70976a9a2794ea57e9458f2bf48b7598e86c4b4e94a60b571e72457a095664f0","sha256:1a5aafecc40e161cc375d6c438e8fd788658bd7aee67fa449edab3cea879aa1f"],"state_sha256":"cd5b040364350225b38d4a760a5cc02d15fe2108fa52574de255d92a1eb4608b"}