{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:QCYJY5K3AGG2T7MPP2FOMOZPZI","short_pith_number":"pith:QCYJY5K3","canonical_record":{"source":{"id":"1507.07957","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-07-28T21:04:57Z","cross_cats_sorted":[],"title_canon_sha256":"154c39642dd552be53812e4361ca5e470835e3a5ebedca1681d1aa7b500cc82d","abstract_canon_sha256":"f32abf5a27cf5a44230d14c6703ea628ed8ec042a066b99f8f80c19770796f60"},"schema_version":"1.0"},"canonical_sha256":"80b09c755b018da9fd8f7e8ae63b2fca072bad501361c72fef9bff6f5bb06f49","source":{"kind":"arxiv","id":"1507.07957","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.07957","created_at":"2026-05-18T01:34:41Z"},{"alias_kind":"arxiv_version","alias_value":"1507.07957v2","created_at":"2026-05-18T01:34:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.07957","created_at":"2026-05-18T01:34:41Z"},{"alias_kind":"pith_short_12","alias_value":"QCYJY5K3AGG2","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"QCYJY5K3AGG2T7MP","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"QCYJY5K3","created_at":"2026-05-18T12:29:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:QCYJY5K3AGG2T7MPP2FOMOZPZI","target":"record","payload":{"canonical_record":{"source":{"id":"1507.07957","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-07-28T21:04:57Z","cross_cats_sorted":[],"title_canon_sha256":"154c39642dd552be53812e4361ca5e470835e3a5ebedca1681d1aa7b500cc82d","abstract_canon_sha256":"f32abf5a27cf5a44230d14c6703ea628ed8ec042a066b99f8f80c19770796f60"},"schema_version":"1.0"},"canonical_sha256":"80b09c755b018da9fd8f7e8ae63b2fca072bad501361c72fef9bff6f5bb06f49","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:34:41.565502Z","signature_b64":"r6AI7oS5C8L2XGza7BximFGgyNrmF6r8uKr5X7Ien1cRqNmEnmb6JrKKec/ZHPF6mkNSkfcA/ek6ruDaK0lfBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"80b09c755b018da9fd8f7e8ae63b2fca072bad501361c72fef9bff6f5bb06f49","last_reissued_at":"2026-05-18T01:34:41.564998Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:34:41.564998Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1507.07957","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":"2026-05-18T01:34:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jnMnIlWb400nD02UV6oa78s1vIK85haVntGhmcVHcm6Uz1SQzoXFqYOeaxC+cd3KEm4FCzV6JIzBCpjBV3mHAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T17:49:06.390340Z"},"content_sha256":"f87c06ca77c7567dd2f7b2f7426927728d233948b9e51db63142d1652744eadc","schema_version":"1.0","event_id":"sha256:f87c06ca77c7567dd2f7b2f7426927728d233948b9e51db63142d1652744eadc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:QCYJY5K3AGG2T7MPP2FOMOZPZI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Focal set of curves in the Minkowski space near lightlike points","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Ana Claudia Nabarro, Andrea de Jesus Sacramento","submitted_at":"2015-07-28T21:04:57Z","abstract_excerpt":"We study the geometry of curves in the Minkowski space and in the de Sitter space, specially at points where the tangent direction is lightlike (i.e. has length zero) called lightlike points of the curve. We define the focal sets of these curves and study the metric structure of them. At the lightlike points, the focal set is not defined. We use singularity theory techniques to carry out our study and investigate the focal set near lightlike points."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.07957","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":"2026-05-18T01:34:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/ry9UxQX7TtR1d0lcgwcE/c+OIUgprRCoW2cK9OWcsU7pCKRik2ptQjQ4I98jKFExJ1i5DaR0kZz2etopoU3Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T17:49:06.390687Z"},"content_sha256":"faea376d24488bf101c335365075dc7a81417bc6c11e40306cd2427dcc53e879","schema_version":"1.0","event_id":"sha256:faea376d24488bf101c335365075dc7a81417bc6c11e40306cd2427dcc53e879"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QCYJY5K3AGG2T7MPP2FOMOZPZI/bundle.json","state_url":"https://pith.science/pith/QCYJY5K3AGG2T7MPP2FOMOZPZI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QCYJY5K3AGG2T7MPP2FOMOZPZI/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":"2026-06-03T17:49:06Z","links":{"resolver":"https://pith.science/pith/QCYJY5K3AGG2T7MPP2FOMOZPZI","bundle":"https://pith.science/pith/QCYJY5K3AGG2T7MPP2FOMOZPZI/bundle.json","state":"https://pith.science/pith/QCYJY5K3AGG2T7MPP2FOMOZPZI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QCYJY5K3AGG2T7MPP2FOMOZPZI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:QCYJY5K3AGG2T7MPP2FOMOZPZI","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":"f32abf5a27cf5a44230d14c6703ea628ed8ec042a066b99f8f80c19770796f60","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-07-28T21:04:57Z","title_canon_sha256":"154c39642dd552be53812e4361ca5e470835e3a5ebedca1681d1aa7b500cc82d"},"schema_version":"1.0","source":{"id":"1507.07957","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.07957","created_at":"2026-05-18T01:34:41Z"},{"alias_kind":"arxiv_version","alias_value":"1507.07957v2","created_at":"2026-05-18T01:34:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.07957","created_at":"2026-05-18T01:34:41Z"},{"alias_kind":"pith_short_12","alias_value":"QCYJY5K3AGG2","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"QCYJY5K3AGG2T7MP","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"QCYJY5K3","created_at":"2026-05-18T12:29:37Z"}],"graph_snapshots":[{"event_id":"sha256:faea376d24488bf101c335365075dc7a81417bc6c11e40306cd2427dcc53e879","target":"graph","created_at":"2026-05-18T01:34:41Z","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 study the geometry of curves in the Minkowski space and in the de Sitter space, specially at points where the tangent direction is lightlike (i.e. has length zero) called lightlike points of the curve. We define the focal sets of these curves and study the metric structure of them. At the lightlike points, the focal set is not defined. We use singularity theory techniques to carry out our study and investigate the focal set near lightlike points.","authors_text":"Ana Claudia Nabarro, Andrea de Jesus Sacramento","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-07-28T21:04:57Z","title":"Focal set of curves in the Minkowski space near lightlike points"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.07957","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:f87c06ca77c7567dd2f7b2f7426927728d233948b9e51db63142d1652744eadc","target":"record","created_at":"2026-05-18T01:34:41Z","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":"f32abf5a27cf5a44230d14c6703ea628ed8ec042a066b99f8f80c19770796f60","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-07-28T21:04:57Z","title_canon_sha256":"154c39642dd552be53812e4361ca5e470835e3a5ebedca1681d1aa7b500cc82d"},"schema_version":"1.0","source":{"id":"1507.07957","kind":"arxiv","version":2}},"canonical_sha256":"80b09c755b018da9fd8f7e8ae63b2fca072bad501361c72fef9bff6f5bb06f49","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"80b09c755b018da9fd8f7e8ae63b2fca072bad501361c72fef9bff6f5bb06f49","first_computed_at":"2026-05-18T01:34:41.564998Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:34:41.564998Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"r6AI7oS5C8L2XGza7BximFGgyNrmF6r8uKr5X7Ien1cRqNmEnmb6JrKKec/ZHPF6mkNSkfcA/ek6ruDaK0lfBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:34:41.565502Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.07957","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f87c06ca77c7567dd2f7b2f7426927728d233948b9e51db63142d1652744eadc","sha256:faea376d24488bf101c335365075dc7a81417bc6c11e40306cd2427dcc53e879"],"state_sha256":"9235125bb54b90f03d3128e8262e959d7373948d5614e66aefde860e1d1c9b42"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"h12AiaxRZ45Mb5y6ENlqyc+Kjtr96oG0OOhTQknQNaYYsCszys7AeiWkcO6vgn3WHE1exILPgw721ppYdHcfCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T17:49:06.392586Z","bundle_sha256":"e83426fa0d33b953afc7576d109d7f1942deefbbb00f6adbf1f70a55516bcb0a"}}