{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:EIABRNNYLNMBX4VAHMFFQOUQXC","short_pith_number":"pith:EIABRNNY","canonical_record":{"source":{"id":"1805.05852","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2018-05-15T15:35:29Z","cross_cats_sorted":["math.GN"],"title_canon_sha256":"b4134c55a40cc4522a2130a975721cde38a94e0ad7674169602177a21b09de84","abstract_canon_sha256":"b66c3ffcd0c38502a734e9609669b86f574dda478b032ce37fb3874d9f5bcc80"},"schema_version":"1.0"},"canonical_sha256":"220018b5b85b581bf2a03b0a583a90b898570999865267db0a16c9b55595a693","source":{"kind":"arxiv","id":"1805.05852","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.05852","created_at":"2026-05-17T23:55:58Z"},{"alias_kind":"arxiv_version","alias_value":"1805.05852v3","created_at":"2026-05-17T23:55:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.05852","created_at":"2026-05-17T23:55:58Z"},{"alias_kind":"pith_short_12","alias_value":"EIABRNNYLNMB","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_16","alias_value":"EIABRNNYLNMBX4VA","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_8","alias_value":"EIABRNNY","created_at":"2026-05-18T12:32:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:EIABRNNYLNMBX4VAHMFFQOUQXC","target":"record","payload":{"canonical_record":{"source":{"id":"1805.05852","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2018-05-15T15:35:29Z","cross_cats_sorted":["math.GN"],"title_canon_sha256":"b4134c55a40cc4522a2130a975721cde38a94e0ad7674169602177a21b09de84","abstract_canon_sha256":"b66c3ffcd0c38502a734e9609669b86f574dda478b032ce37fb3874d9f5bcc80"},"schema_version":"1.0"},"canonical_sha256":"220018b5b85b581bf2a03b0a583a90b898570999865267db0a16c9b55595a693","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:55:58.753906Z","signature_b64":"FzfJ38Cmq7cCkfMXU2nIYi4jYX4COLrbXDR3cNlw7bGuZZYnTojp2qeB6gfBHV2hyizTaKw38Ndzz2aQ0LkYAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"220018b5b85b581bf2a03b0a583a90b898570999865267db0a16c9b55595a693","last_reissued_at":"2026-05-17T23:55:58.753223Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:55:58.753223Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1805.05852","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:55:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eR64xwSKdjTUsxgH6BSKCkmJimH4Hph7qH2bD08nrXK9znF0CDtu9YwyiyvKc1x+t1B5oGwGwBhkZwgvG3XEAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-24T15:56:45.921232Z"},"content_sha256":"40775d2ee0543f34abf3a3d0f39c539587476e1b25368392205b4f5fb64f5bdc","schema_version":"1.0","event_id":"sha256:40775d2ee0543f34abf3a3d0f39c539587476e1b25368392205b4f5fb64f5bdc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:EIABRNNYLNMBX4VAHMFFQOUQXC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Lifts of spherical Morse functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GN"],"primary_cat":"math.GT","authors_text":"Naoki Kitazawa","submitted_at":"2018-05-15T15:35:29Z","abstract_excerpt":"In studies of smooth maps with good differential topological conditions such as immersions, embeddings, Morse functions and their higher dimensional versions including fold maps and application to geometry, especially algebraic and differential topology of manifolds, liftings or desingulizations of maps of appropriate classes to other maps of other appropriate classes are fundamental and important. In this paper, we consider Morse functions such that inverse images of regular values are disjoint unions of spheres, which are extensions of Morse functions with just two singular points on homotop"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.05852","kind":"arxiv","version":3},"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":"2026-05-17T23:55:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"L639ksRoaS6puTPvcYYfljf130g+V/N3ld7Fa8s5LZGbhjUjLB36rklJ9rBV1hxFdxKSH5dfOw9EAiO1GqKCDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-24T15:56:45.921927Z"},"content_sha256":"0bf8bc8dbedc1b1d9a72a0b82bce54df55fb9003578bcb7b19a1fa98af0727ff","schema_version":"1.0","event_id":"sha256:0bf8bc8dbedc1b1d9a72a0b82bce54df55fb9003578bcb7b19a1fa98af0727ff"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EIABRNNYLNMBX4VAHMFFQOUQXC/bundle.json","state_url":"https://pith.science/pith/EIABRNNYLNMBX4VAHMFFQOUQXC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EIABRNNYLNMBX4VAHMFFQOUQXC/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":"2026-05-24T15:56:45Z","links":{"resolver":"https://pith.science/pith/EIABRNNYLNMBX4VAHMFFQOUQXC","bundle":"https://pith.science/pith/EIABRNNYLNMBX4VAHMFFQOUQXC/bundle.json","state":"https://pith.science/pith/EIABRNNYLNMBX4VAHMFFQOUQXC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EIABRNNYLNMBX4VAHMFFQOUQXC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:EIABRNNYLNMBX4VAHMFFQOUQXC","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":"b66c3ffcd0c38502a734e9609669b86f574dda478b032ce37fb3874d9f5bcc80","cross_cats_sorted":["math.GN"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2018-05-15T15:35:29Z","title_canon_sha256":"b4134c55a40cc4522a2130a975721cde38a94e0ad7674169602177a21b09de84"},"schema_version":"1.0","source":{"id":"1805.05852","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.05852","created_at":"2026-05-17T23:55:58Z"},{"alias_kind":"arxiv_version","alias_value":"1805.05852v3","created_at":"2026-05-17T23:55:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.05852","created_at":"2026-05-17T23:55:58Z"},{"alias_kind":"pith_short_12","alias_value":"EIABRNNYLNMB","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_16","alias_value":"EIABRNNYLNMBX4VA","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_8","alias_value":"EIABRNNY","created_at":"2026-05-18T12:32:22Z"}],"graph_snapshots":[{"event_id":"sha256:0bf8bc8dbedc1b1d9a72a0b82bce54df55fb9003578bcb7b19a1fa98af0727ff","target":"graph","created_at":"2026-05-17T23:55:58Z","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":"In studies of smooth maps with good differential topological conditions such as immersions, embeddings, Morse functions and their higher dimensional versions including fold maps and application to geometry, especially algebraic and differential topology of manifolds, liftings or desingulizations of maps of appropriate classes to other maps of other appropriate classes are fundamental and important. In this paper, we consider Morse functions such that inverse images of regular values are disjoint unions of spheres, which are extensions of Morse functions with just two singular points on homotop","authors_text":"Naoki Kitazawa","cross_cats":["math.GN"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2018-05-15T15:35:29Z","title":"Lifts of spherical Morse functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.05852","kind":"arxiv","version":3},"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":"sha256:40775d2ee0543f34abf3a3d0f39c539587476e1b25368392205b4f5fb64f5bdc","target":"record","created_at":"2026-05-17T23:55:58Z","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":"b66c3ffcd0c38502a734e9609669b86f574dda478b032ce37fb3874d9f5bcc80","cross_cats_sorted":["math.GN"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2018-05-15T15:35:29Z","title_canon_sha256":"b4134c55a40cc4522a2130a975721cde38a94e0ad7674169602177a21b09de84"},"schema_version":"1.0","source":{"id":"1805.05852","kind":"arxiv","version":3}},"canonical_sha256":"220018b5b85b581bf2a03b0a583a90b898570999865267db0a16c9b55595a693","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"220018b5b85b581bf2a03b0a583a90b898570999865267db0a16c9b55595a693","first_computed_at":"2026-05-17T23:55:58.753223Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:55:58.753223Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FzfJ38Cmq7cCkfMXU2nIYi4jYX4COLrbXDR3cNlw7bGuZZYnTojp2qeB6gfBHV2hyizTaKw38Ndzz2aQ0LkYAA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:55:58.753906Z","signed_message":"canonical_sha256_bytes"},"source_id":"1805.05852","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:40775d2ee0543f34abf3a3d0f39c539587476e1b25368392205b4f5fb64f5bdc","sha256:0bf8bc8dbedc1b1d9a72a0b82bce54df55fb9003578bcb7b19a1fa98af0727ff"],"state_sha256":"91863fc40c84e45b6446694d6913f7f0d3077e74cf3425ac630d2de024143825"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"V95X++AFkfnbrROqW6zOwtpnLki/o/xL3vcY5pJ96FGw7el5nI6VbdOilhHyWMLz+MYMiXIzmUtbOTy0dVM5Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-24T15:56:45.925396Z","bundle_sha256":"4c56b7aaf2e97e08f2619c50c946f5ed7f3291f3464f302d4eebb53b570d39d0"}}