{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:VBQP3TFRCJ535OIPZSXYQUNKAW","short_pith_number":"pith:VBQP3TFR","canonical_record":{"source":{"id":"1104.4562","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-04-23T14:23:14Z","cross_cats_sorted":[],"title_canon_sha256":"a1666ee61d585635515c865b33620e9a405e94668289d5f96d6652414c3e507a","abstract_canon_sha256":"36c41b75aa6938d59bd0aa4c6b75741fe6b17ea22105b4716ec595ddfaa69126"},"schema_version":"1.0"},"canonical_sha256":"a860fdccb1127bbeb90fccaf8851aa05afe26c32ef7f18ac194e94a86c1889db","source":{"kind":"arxiv","id":"1104.4562","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"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:VBQP3TFRCJ535OIPZSXYQUNKAW","target":"record","payload":{"canonical_record":{"source":{"id":"1104.4562","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-04-23T14:23:14Z","cross_cats_sorted":[],"title_canon_sha256":"a1666ee61d585635515c865b33620e9a405e94668289d5f96d6652414c3e507a","abstract_canon_sha256":"36c41b75aa6938d59bd0aa4c6b75741fe6b17ea22105b4716ec595ddfaa69126"},"schema_version":"1.0"},"canonical_sha256":"a860fdccb1127bbeb90fccaf8851aa05afe26c32ef7f18ac194e94a86c1889db","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:23:33.870098Z","signature_b64":"h2uLrSAEXKbJE+mEmzqtlph3ZLEAodEfu9q2DEpTfjzSQuy4D2ldWouglVP3CFIAcVqZxVuTNMGV2St53iApBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a860fdccb1127bbeb90fccaf8851aa05afe26c32ef7f18ac194e94a86c1889db","last_reissued_at":"2026-05-18T04:23:33.869680Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:23:33.869680Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1104.4562","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:23:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VikUXECwsda5s+oQovgKVTqj7mvEoNSKp9M9/DpTw6/enWAG5WtyEx1zCvErD5FRQ/Bzvcr2SYITCfZFm2sDCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T00:41:10.538031Z"},"content_sha256":"262e3c4cbab6fa8e1751bed4ae6e174b025fc210fa63be6c4e27d53d67f12c2e","schema_version":"1.0","event_id":"sha256:262e3c4cbab6fa8e1751bed4ae6e174b025fc210fa63be6c4e27d53d67f12c2e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:VBQP3TFRCJ535OIPZSXYQUNKAW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Semi-linear Torsional Rigidity on a Complete Riemannian Two-Manifold","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Jie Xiao","submitted_at":"2011-04-23T14:23:14Z","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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.4562","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:23:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"12HNgZmoNHGPPBiqzyHD1AACXPg4QG7SEEfDMb1TC7vbcVSKwL+TlX518sGmPce0MZHRt1W8wrk+gzTt7mIDDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T00:41:10.538389Z"},"content_sha256":"b8cbded688039017862e480833577b1230f03b8e8cc7df6e3fd44591faae54dd","schema_version":"1.0","event_id":"sha256:b8cbded688039017862e480833577b1230f03b8e8cc7df6e3fd44591faae54dd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VBQP3TFRCJ535OIPZSXYQUNKAW/bundle.json","state_url":"https://pith.science/pith/VBQP3TFRCJ535OIPZSXYQUNKAW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VBQP3TFRCJ535OIPZSXYQUNKAW/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-02T00:41:10Z","links":{"resolver":"https://pith.science/pith/VBQP3TFRCJ535OIPZSXYQUNKAW","bundle":"https://pith.science/pith/VBQP3TFRCJ535OIPZSXYQUNKAW/bundle.json","state":"https://pith.science/pith/VBQP3TFRCJ535OIPZSXYQUNKAW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VBQP3TFRCJ535OIPZSXYQUNKAW/bundle.json"},"state":{"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"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cIpHf3aBzMyR+D4BnK8DhAXEnGoopsmRe0HHS1TdsWehwMwA8tocM77iVC582mdC4R0xxcvaVSkpNlj7s/6MCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T00:41:10.540351Z","bundle_sha256":"0bc5cd252da665bf7b8531dc020e01ef90e3b0219e415e6972683ec8e159ccc9"}}