{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:N6QHM4QEMDLE4ZWZJG7JVUCKDJ","short_pith_number":"pith:N6QHM4QE","canonical_record":{"source":{"id":"1105.0553","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-05-03T11:33:37Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"8c2cae515cf15807b66de991514eed7458b4cba6ad372ad9497a812dfcd733b5","abstract_canon_sha256":"8ee90bec6e51e7db7324d87847ceac8f57c439e5346a4d2d3a63a6962aa91ec6"},"schema_version":"1.0"},"canonical_sha256":"6fa076720460d64e66d949be9ad04a1a5a7ffa7318e321f0898170c9628e3b0a","source":{"kind":"arxiv","id":"1105.0553","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1105.0553","created_at":"2026-05-18T04:20:12Z"},{"alias_kind":"arxiv_version","alias_value":"1105.0553v2","created_at":"2026-05-18T04:20:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.0553","created_at":"2026-05-18T04:20:12Z"},{"alias_kind":"pith_short_12","alias_value":"N6QHM4QEMDLE","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_16","alias_value":"N6QHM4QEMDLE4ZWZ","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_8","alias_value":"N6QHM4QE","created_at":"2026-05-18T12:26:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:N6QHM4QEMDLE4ZWZJG7JVUCKDJ","target":"record","payload":{"canonical_record":{"source":{"id":"1105.0553","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-05-03T11:33:37Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"8c2cae515cf15807b66de991514eed7458b4cba6ad372ad9497a812dfcd733b5","abstract_canon_sha256":"8ee90bec6e51e7db7324d87847ceac8f57c439e5346a4d2d3a63a6962aa91ec6"},"schema_version":"1.0"},"canonical_sha256":"6fa076720460d64e66d949be9ad04a1a5a7ffa7318e321f0898170c9628e3b0a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:20:12.746143Z","signature_b64":"ICrBNeDztj0Tn1cZR1EIDCqomAqPMJlp4dRHPv6Pkq13wNVHUDAmyveU+RY3IwEBkZrgaLfbQSM8SSLwdDh9Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6fa076720460d64e66d949be9ad04a1a5a7ffa7318e321f0898170c9628e3b0a","last_reissued_at":"2026-05-18T04:20:12.745583Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:20:12.745583Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1105.0553","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:20:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4AF3J7zbD9EgKBo/kouINxn0JI+20CShrk+GXqeAn0e/b0UzwqTXtQXPFHMHWz8c7B72kjg5Iw0y9DYt74gfDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T21:54:16.158507Z"},"content_sha256":"ed194e7cab55bbec2122c2c893cf2d4e95fccd3cfe1203390af87d472914a082","schema_version":"1.0","event_id":"sha256:ed194e7cab55bbec2122c2c893cf2d4e95fccd3cfe1203390af87d472914a082"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:N6QHM4QEMDLE4ZWZJG7JVUCKDJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Liouville's equation for curvature and systolic defect","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Mikhail Katz","submitted_at":"2011-05-03T11:33:37Z","abstract_excerpt":"We analyze the probabilistic variance of a solution of Liouville's equation for curvature, given suitable bounds on the Gaussian curvature. The related systolic geometry was recently studied by Horowitz, Katz, and Katz, where we obtained a strengthening of Loewner's torus inequality containing a \"defect term\", similar to Bonnesen's strengthening of the isoperimetric inequality. Here the analogous isosystolic defect term depends on the metric and \"measures\" its deviation from being flat. Namely, the defect is the variance of the function f which appears as the conformal factor expressing the me"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.0553","kind":"arxiv","version":2},"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":"2026-05-18T04:20:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YT/ZrR9GaPsYkxxpaNQXxE69YsFup89LYIC95FLYCbBSZoMJUeuCvXeLNDDB6lZYRlf+SYTzjn0gpQZ0ROPZCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T21:54:16.159146Z"},"content_sha256":"a53589cb793a77f97d2a266a34633467df0d92096e30fadf02a70754a503398d","schema_version":"1.0","event_id":"sha256:a53589cb793a77f97d2a266a34633467df0d92096e30fadf02a70754a503398d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/N6QHM4QEMDLE4ZWZJG7JVUCKDJ/bundle.json","state_url":"https://pith.science/pith/N6QHM4QEMDLE4ZWZJG7JVUCKDJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/N6QHM4QEMDLE4ZWZJG7JVUCKDJ/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":"2026-06-22T21:54:16Z","links":{"resolver":"https://pith.science/pith/N6QHM4QEMDLE4ZWZJG7JVUCKDJ","bundle":"https://pith.science/pith/N6QHM4QEMDLE4ZWZJG7JVUCKDJ/bundle.json","state":"https://pith.science/pith/N6QHM4QEMDLE4ZWZJG7JVUCKDJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/N6QHM4QEMDLE4ZWZJG7JVUCKDJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:N6QHM4QEMDLE4ZWZJG7JVUCKDJ","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":"8ee90bec6e51e7db7324d87847ceac8f57c439e5346a4d2d3a63a6962aa91ec6","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-05-03T11:33:37Z","title_canon_sha256":"8c2cae515cf15807b66de991514eed7458b4cba6ad372ad9497a812dfcd733b5"},"schema_version":"1.0","source":{"id":"1105.0553","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1105.0553","created_at":"2026-05-18T04:20:12Z"},{"alias_kind":"arxiv_version","alias_value":"1105.0553v2","created_at":"2026-05-18T04:20:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.0553","created_at":"2026-05-18T04:20:12Z"},{"alias_kind":"pith_short_12","alias_value":"N6QHM4QEMDLE","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_16","alias_value":"N6QHM4QEMDLE4ZWZ","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_8","alias_value":"N6QHM4QE","created_at":"2026-05-18T12:26:37Z"}],"graph_snapshots":[{"event_id":"sha256:a53589cb793a77f97d2a266a34633467df0d92096e30fadf02a70754a503398d","target":"graph","created_at":"2026-05-18T04:20:12Z","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":"We analyze the probabilistic variance of a solution of Liouville's equation for curvature, given suitable bounds on the Gaussian curvature. The related systolic geometry was recently studied by Horowitz, Katz, and Katz, where we obtained a strengthening of Loewner's torus inequality containing a \"defect term\", similar to Bonnesen's strengthening of the isoperimetric inequality. Here the analogous isosystolic defect term depends on the metric and \"measures\" its deviation from being flat. Namely, the defect is the variance of the function f which appears as the conformal factor expressing the me","authors_text":"Mikhail Katz","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-05-03T11:33:37Z","title":"Liouville's equation for curvature and systolic defect"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.0553","kind":"arxiv","version":2},"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:ed194e7cab55bbec2122c2c893cf2d4e95fccd3cfe1203390af87d472914a082","target":"record","created_at":"2026-05-18T04:20:12Z","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":"8ee90bec6e51e7db7324d87847ceac8f57c439e5346a4d2d3a63a6962aa91ec6","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-05-03T11:33:37Z","title_canon_sha256":"8c2cae515cf15807b66de991514eed7458b4cba6ad372ad9497a812dfcd733b5"},"schema_version":"1.0","source":{"id":"1105.0553","kind":"arxiv","version":2}},"canonical_sha256":"6fa076720460d64e66d949be9ad04a1a5a7ffa7318e321f0898170c9628e3b0a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6fa076720460d64e66d949be9ad04a1a5a7ffa7318e321f0898170c9628e3b0a","first_computed_at":"2026-05-18T04:20:12.745583Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:20:12.745583Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ICrBNeDztj0Tn1cZR1EIDCqomAqPMJlp4dRHPv6Pkq13wNVHUDAmyveU+RY3IwEBkZrgaLfbQSM8SSLwdDh9Aw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:20:12.746143Z","signed_message":"canonical_sha256_bytes"},"source_id":"1105.0553","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ed194e7cab55bbec2122c2c893cf2d4e95fccd3cfe1203390af87d472914a082","sha256:a53589cb793a77f97d2a266a34633467df0d92096e30fadf02a70754a503398d"],"state_sha256":"86bd081264f80a2fd25ecaead3455d5d89edeff5ff67da4e44814db7916952da"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Oja7tD+ii4jwCiKX2KOYjER7dM4NiTw6c7sUHg8LqHK5gkCk9D1FW4gsRLG+XagcQyhvcsBcLbFKM18glSqmAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T21:54:16.161786Z","bundle_sha256":"1214167d702e425d49240b432c54c8e056f8eb77ad6896d5aedb90bf7a0950aa"}}