{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:DA3NXLBJ2TRET3MDXZMOLI777E","short_pith_number":"pith:DA3NXLBJ","canonical_record":{"source":{"id":"1308.2136","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-08-09T14:31:30Z","cross_cats_sorted":[],"title_canon_sha256":"83134e3389e77933c6b6fe2c5ca369194f19b56b5ba6ebe458f1fa7dcd7125a5","abstract_canon_sha256":"0b5e34b76ddb504fa198047c27bcb71291c7a4b0ff241a1eb7fe1bf2b49eecda"},"schema_version":"1.0"},"canonical_sha256":"1836dbac29d4e249ed83be58e5a3fff914f90623094214580cc4aa386eda66b4","source":{"kind":"arxiv","id":"1308.2136","version":6},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.2136","created_at":"2026-05-18T01:31:12Z"},{"alias_kind":"arxiv_version","alias_value":"1308.2136v6","created_at":"2026-05-18T01:31:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.2136","created_at":"2026-05-18T01:31:12Z"},{"alias_kind":"pith_short_12","alias_value":"DA3NXLBJ2TRE","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_16","alias_value":"DA3NXLBJ2TRET3MD","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_8","alias_value":"DA3NXLBJ","created_at":"2026-05-18T12:27:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:DA3NXLBJ2TRET3MDXZMOLI777E","target":"record","payload":{"canonical_record":{"source":{"id":"1308.2136","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-08-09T14:31:30Z","cross_cats_sorted":[],"title_canon_sha256":"83134e3389e77933c6b6fe2c5ca369194f19b56b5ba6ebe458f1fa7dcd7125a5","abstract_canon_sha256":"0b5e34b76ddb504fa198047c27bcb71291c7a4b0ff241a1eb7fe1bf2b49eecda"},"schema_version":"1.0"},"canonical_sha256":"1836dbac29d4e249ed83be58e5a3fff914f90623094214580cc4aa386eda66b4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:31:12.007350Z","signature_b64":"laLtdQLez/gZEmKrhZ4ibgjZ1TLIu0CSAS9A7+XzYm82vjxUlt7fIhUPjSb6hzHo5lJgmeQBHHsUwQHh9dCCAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1836dbac29d4e249ed83be58e5a3fff914f90623094214580cc4aa386eda66b4","last_reissued_at":"2026-05-18T01:31:12.006712Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:31:12.006712Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1308.2136","source_version":6,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:31:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FpHvdh2nPoZSLT6mvnzKvbNPxWWturgRoOAsq33WOoMzTCJjq4Daq2Hq+t/aVDdG6wUXq7p/YESaVboctwPQBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T06:50:09.041823Z"},"content_sha256":"8f23960b5348f200d9a4237408aa3094ae4e3f1e995f4227feaf2347f4e8e999","schema_version":"1.0","event_id":"sha256:8f23960b5348f200d9a4237408aa3094ae4e3f1e995f4227feaf2347f4e8e999"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:DA3NXLBJ2TRET3MDXZMOLI777E","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Behavior of Gaussian curvature and mean curvature near non-degenerate singular points on wave fronts","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Kentaro Saji, Kotaro Yamada, Luciana F. Martins, Masaaki Umehara","submitted_at":"2013-08-09T14:31:30Z","abstract_excerpt":"We define cuspidal curvature $\\kappa_c$ (resp. normalized cuspidal curvature $\\mu_c$) along cuspidal edges (resp. at swallowtail singularity) in Riemannian $3$-manifolds, and show that it gives a coefficient of the divergent term of the mean curvature function. Moreover, we show that the product $\\kappa_\\Pi$ called the product curvature (resp. $\\mu_\\Pi$ called normalized product curvature) of $\\kappa_c$ (resp. $\\mu_c$) and the limiting normal curvature $\\kappa_\\nu$ is an intrinsic invariant of the surface, and is closely related to the boundedness of the Gaussian curvature. We also consider th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.2136","kind":"arxiv","version":6},"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":"2026-05-18T01:31:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cNalxQN+zyWqZ1SyYylUSuzfs/Dxg8GlqDOcCX5HwcVZMrhbwsr4mNorSgN13DNmHJXcgeqI2Bp2PPBXvH2cDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T06:50:09.042528Z"},"content_sha256":"101e4edbd7591c0d8a84d71c2ccdf007705ac0c9a84e90799ae7614f020ec49c","schema_version":"1.0","event_id":"sha256:101e4edbd7591c0d8a84d71c2ccdf007705ac0c9a84e90799ae7614f020ec49c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DA3NXLBJ2TRET3MDXZMOLI777E/bundle.json","state_url":"https://pith.science/pith/DA3NXLBJ2TRET3MDXZMOLI777E/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DA3NXLBJ2TRET3MDXZMOLI777E/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":"2026-05-26T06:50:09Z","links":{"resolver":"https://pith.science/pith/DA3NXLBJ2TRET3MDXZMOLI777E","bundle":"https://pith.science/pith/DA3NXLBJ2TRET3MDXZMOLI777E/bundle.json","state":"https://pith.science/pith/DA3NXLBJ2TRET3MDXZMOLI777E/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DA3NXLBJ2TRET3MDXZMOLI777E/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:DA3NXLBJ2TRET3MDXZMOLI777E","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":"0b5e34b76ddb504fa198047c27bcb71291c7a4b0ff241a1eb7fe1bf2b49eecda","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-08-09T14:31:30Z","title_canon_sha256":"83134e3389e77933c6b6fe2c5ca369194f19b56b5ba6ebe458f1fa7dcd7125a5"},"schema_version":"1.0","source":{"id":"1308.2136","kind":"arxiv","version":6}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.2136","created_at":"2026-05-18T01:31:12Z"},{"alias_kind":"arxiv_version","alias_value":"1308.2136v6","created_at":"2026-05-18T01:31:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.2136","created_at":"2026-05-18T01:31:12Z"},{"alias_kind":"pith_short_12","alias_value":"DA3NXLBJ2TRE","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_16","alias_value":"DA3NXLBJ2TRET3MD","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_8","alias_value":"DA3NXLBJ","created_at":"2026-05-18T12:27:40Z"}],"graph_snapshots":[{"event_id":"sha256:101e4edbd7591c0d8a84d71c2ccdf007705ac0c9a84e90799ae7614f020ec49c","target":"graph","created_at":"2026-05-18T01:31:12Z","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 define cuspidal curvature $\\kappa_c$ (resp. normalized cuspidal curvature $\\mu_c$) along cuspidal edges (resp. at swallowtail singularity) in Riemannian $3$-manifolds, and show that it gives a coefficient of the divergent term of the mean curvature function. Moreover, we show that the product $\\kappa_\\Pi$ called the product curvature (resp. $\\mu_\\Pi$ called normalized product curvature) of $\\kappa_c$ (resp. $\\mu_c$) and the limiting normal curvature $\\kappa_\\nu$ is an intrinsic invariant of the surface, and is closely related to the boundedness of the Gaussian curvature. We also consider th","authors_text":"Kentaro Saji, Kotaro Yamada, Luciana F. Martins, Masaaki Umehara","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-08-09T14:31:30Z","title":"Behavior of Gaussian curvature and mean curvature near non-degenerate singular points on wave fronts"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.2136","kind":"arxiv","version":6},"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":"sha256:8f23960b5348f200d9a4237408aa3094ae4e3f1e995f4227feaf2347f4e8e999","target":"record","created_at":"2026-05-18T01:31:12Z","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":"0b5e34b76ddb504fa198047c27bcb71291c7a4b0ff241a1eb7fe1bf2b49eecda","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-08-09T14:31:30Z","title_canon_sha256":"83134e3389e77933c6b6fe2c5ca369194f19b56b5ba6ebe458f1fa7dcd7125a5"},"schema_version":"1.0","source":{"id":"1308.2136","kind":"arxiv","version":6}},"canonical_sha256":"1836dbac29d4e249ed83be58e5a3fff914f90623094214580cc4aa386eda66b4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1836dbac29d4e249ed83be58e5a3fff914f90623094214580cc4aa386eda66b4","first_computed_at":"2026-05-18T01:31:12.006712Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:31:12.006712Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"laLtdQLez/gZEmKrhZ4ibgjZ1TLIu0CSAS9A7+XzYm82vjxUlt7fIhUPjSb6hzHo5lJgmeQBHHsUwQHh9dCCAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:31:12.007350Z","signed_message":"canonical_sha256_bytes"},"source_id":"1308.2136","source_kind":"arxiv","source_version":6}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8f23960b5348f200d9a4237408aa3094ae4e3f1e995f4227feaf2347f4e8e999","sha256:101e4edbd7591c0d8a84d71c2ccdf007705ac0c9a84e90799ae7614f020ec49c"],"state_sha256":"4f6cfc285dd61ec6ccc1d1645ceb760bfce47c98fd617c173e2deb39c3b14a38"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vlLV6N3jQTrBaqkPwXLlI7mvqDS1rpyfRVhRDOCkvN9gbe9DI5KZesgn8zgO4tNsF7DWrGkDE1CIiZaeGmlNBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T06:50:09.046351Z","bundle_sha256":"078378514c4e9273c7c5525321a6812967502dba6795bf0a78a3843b89a142bf"}}