{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:FXLRZ6HLWGYBBSFGDTLBZ2GVIH","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":"bbdbce31327bd384fe2c5c4251705cc7f534befaa8442204ee2ee04e83eecf3d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2024-09-10T00:01:44Z","title_canon_sha256":"4cc500075d017e74ee76073961687db876b03f947c3639bbebe0a0404f0c4065"},"schema_version":"1.0","source":{"id":"2409.06117","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2409.06117","created_at":"2026-07-05T11:05:34Z"},{"alias_kind":"arxiv_version","alias_value":"2409.06117v5","created_at":"2026-07-05T11:05:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2409.06117","created_at":"2026-07-05T11:05:34Z"},{"alias_kind":"pith_short_12","alias_value":"FXLRZ6HLWGYB","created_at":"2026-07-05T11:05:34Z"},{"alias_kind":"pith_short_16","alias_value":"FXLRZ6HLWGYBBSFG","created_at":"2026-07-05T11:05:34Z"},{"alias_kind":"pith_short_8","alias_value":"FXLRZ6HL","created_at":"2026-07-05T11:05:34Z"}],"graph_snapshots":[{"event_id":"sha256:5fdb8d9ef321338f110c1e864143fe92980928f2bbf900870f47ce7b3e828504","target":"graph","created_at":"2026-07-05T11:05:34Z","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/2409.06117/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we obtain that the logarithmic Sobolev and $\\mathcal{W}$-functionals admit remarkable power series expansions when appropriate test functions are selected. Using these expansions formulas, we prove that for an open subset $V$ in an $n$-dimensional manifold $M$ with $\\bar{V}\\subset M$ satisfying: (a)The scalar curvature of $V$ satisfies the lower bound:$$\\operatorname{Sc}(x) \\geq n(n-1)K \\quad \\text{for all } x \\in V,$$ (b) The isoperimetric profile of $V$ is no less than that of space form $M^n_K$:$$ \\operatorname{I}(V,\\beta) := \\inf_{\\substack{\\Omega\\subset V \\\\ \\mathrm{Vol}(\\O","authors_text":"Liang Cheng","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2024-09-10T00:01:44Z","title":"The power series expansions of logarithmic Sobolev, $\\mathcal{W}$- functionals and scalar curvature rigidity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.06117","kind":"arxiv","version":5},"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:fc2c7377eea0ab63114e7c4ea1f3116494fcefd24d04bbbd71ef7edb0c26edae","target":"record","created_at":"2026-07-05T11:05:34Z","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":"bbdbce31327bd384fe2c5c4251705cc7f534befaa8442204ee2ee04e83eecf3d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2024-09-10T00:01:44Z","title_canon_sha256":"4cc500075d017e74ee76073961687db876b03f947c3639bbebe0a0404f0c4065"},"schema_version":"1.0","source":{"id":"2409.06117","kind":"arxiv","version":5}},"canonical_sha256":"2dd71cf8ebb1b010c8a61cd61ce8d541d24e804ff05e0ab68de8d72b3e8517e0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2dd71cf8ebb1b010c8a61cd61ce8d541d24e804ff05e0ab68de8d72b3e8517e0","first_computed_at":"2026-07-05T11:05:34.717892Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:05:34.717892Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xc/FcTo5xJSschHtotELA1ng7zLXLxNFjB+lvMRT4+M+oK1cZS2MyJEZtkDxZNMExG8wdzcq85qN0qh0fEjlAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T11:05:34.718354Z","signed_message":"canonical_sha256_bytes"},"source_id":"2409.06117","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fc2c7377eea0ab63114e7c4ea1f3116494fcefd24d04bbbd71ef7edb0c26edae","sha256:5fdb8d9ef321338f110c1e864143fe92980928f2bbf900870f47ce7b3e828504"],"state_sha256":"2e7874191916fefd98c71376baf8ac9a53cb0f4c2aafbe2a1b24000c83846dc1"}