{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:VBQP3TFRCJ535OIPZSXYQUNKAW","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":"36c41b75aa6938d59bd0aa4c6b75741fe6b17ea22105b4716ec595ddfaa69126","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-04-23T14:23:14Z","title_canon_sha256":"a1666ee61d585635515c865b33620e9a405e94668289d5f96d6652414c3e507a"},"schema_version":"1.0","source":{"id":"1104.4562","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1104.4562","created_at":"2026-05-18T04:23:33Z"},{"alias_kind":"arxiv_version","alias_value":"1104.4562v1","created_at":"2026-05-18T04:23:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.4562","created_at":"2026-05-18T04:23:33Z"},{"alias_kind":"pith_short_12","alias_value":"VBQP3TFRCJ53","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_16","alias_value":"VBQP3TFRCJ535OIP","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_8","alias_value":"VBQP3TFR","created_at":"2026-05-18T12:26:42Z"}],"graph_snapshots":[{"event_id":"sha256:b8cbded688039017862e480833577b1230f03b8e8cc7df6e3fd44591faae54dd","target":"graph","created_at":"2026-05-18T04:23:33Z","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":"This note is concerned with some essential properties (optimal isoperimetry, first variation, and monotonicity formula) of the so-called $[0,1)\\ni\\gamma$-torsional rigidity $\\mathcal{T}_{\\gamma,\\mathsf{g}}$ on a complete Riemannian two-manifold $(\\mathbb M^2,\\mathsf{g})$. Even in the special case of $\\mathbb R^2$, major results are new.","authors_text":"Jie Xiao","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-04-23T14:23:14Z","title":"The Semi-linear Torsional Rigidity on a Complete Riemannian Two-Manifold"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.4562","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":"sha256:262e3c4cbab6fa8e1751bed4ae6e174b025fc210fa63be6c4e27d53d67f12c2e","target":"record","created_at":"2026-05-18T04:23:33Z","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":"36c41b75aa6938d59bd0aa4c6b75741fe6b17ea22105b4716ec595ddfaa69126","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-04-23T14:23:14Z","title_canon_sha256":"a1666ee61d585635515c865b33620e9a405e94668289d5f96d6652414c3e507a"},"schema_version":"1.0","source":{"id":"1104.4562","kind":"arxiv","version":1}},"canonical_sha256":"a860fdccb1127bbeb90fccaf8851aa05afe26c32ef7f18ac194e94a86c1889db","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a860fdccb1127bbeb90fccaf8851aa05afe26c32ef7f18ac194e94a86c1889db","first_computed_at":"2026-05-18T04:23:33.869680Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:23:33.869680Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"h2uLrSAEXKbJE+mEmzqtlph3ZLEAodEfu9q2DEpTfjzSQuy4D2ldWouglVP3CFIAcVqZxVuTNMGV2St53iApBg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:23:33.870098Z","signed_message":"canonical_sha256_bytes"},"source_id":"1104.4562","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:262e3c4cbab6fa8e1751bed4ae6e174b025fc210fa63be6c4e27d53d67f12c2e","sha256:b8cbded688039017862e480833577b1230f03b8e8cc7df6e3fd44591faae54dd"],"state_sha256":"39669455e92c643d12b83152ab74839595dcaaadc98dce35f562e42620eb591e"}