{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:ECDWIBYPJIV5BSLGG2J5A44GDK","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":"84e1f3be68c85001f75589b97fc74d473b018dd4a78fa7336cc4341a0ac73e0b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2019-02-11T19:16:05Z","title_canon_sha256":"486e5f64a38b76cd16e6a5959f3e0140b5832b3e6def2825e7bc7a676f39b17e"},"schema_version":"1.0","source":{"id":"1902.04101","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1902.04101","created_at":"2026-05-17T23:51:04Z"},{"alias_kind":"arxiv_version","alias_value":"1902.04101v2","created_at":"2026-05-17T23:51:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1902.04101","created_at":"2026-05-17T23:51:04Z"},{"alias_kind":"pith_short_12","alias_value":"ECDWIBYPJIV5","created_at":"2026-05-18T12:33:15Z"},{"alias_kind":"pith_short_16","alias_value":"ECDWIBYPJIV5BSLG","created_at":"2026-05-18T12:33:15Z"},{"alias_kind":"pith_short_8","alias_value":"ECDWIBYP","created_at":"2026-05-18T12:33:15Z"}],"graph_snapshots":[{"event_id":"sha256:a07a380b2527c8a81edbd23c69214aac11b21ec6fc1e7155b44907634dac5700","target":"graph","created_at":"2026-05-17T23:51:04Z","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":"(Co)bordisms of manifolds and maps are fundamental and important objects in algebraic and differential topology of manifolds and related studies were started by Thom etc.. Cobordisms of Morse functions were introduced and have been studied as a branch of the algebraic and differential topological theory of Morse functions and their higher dimensional versions, or the global singularity theory, by Kalm\\'{a}r, Ikegami, Sadykov, Saeki, Wrazidlo, Yamamoto etc. since 2000s.\n  Cobordism relations are in most cases defined as the following for example; two closed manifolds of a fixed dimension or map","authors_text":"Naoki Kitazawa","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2019-02-11T19:16:05Z","title":"On defining products of cobordism classes of Morse functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.04101","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":"sha256:2b94cdbb4366fc40c5b16b2d23695dfb5a61e9256660251c11ae453b7f27347d","target":"record","created_at":"2026-05-17T23:51:04Z","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":"84e1f3be68c85001f75589b97fc74d473b018dd4a78fa7336cc4341a0ac73e0b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2019-02-11T19:16:05Z","title_canon_sha256":"486e5f64a38b76cd16e6a5959f3e0140b5832b3e6def2825e7bc7a676f39b17e"},"schema_version":"1.0","source":{"id":"1902.04101","kind":"arxiv","version":2}},"canonical_sha256":"208764070f4a2bd0c9663693d073861ab07c9e6d167a15959548dde94e041e82","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"208764070f4a2bd0c9663693d073861ab07c9e6d167a15959548dde94e041e82","first_computed_at":"2026-05-17T23:51:04.986160Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:51:04.986160Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bZEYz+s34xCTZA3l2g9u/U+5m70vWdO4ARarblhl54xwW+DhiYkseFdPLDigSTLh0+ZJuagNcRuJ589aNtwpCg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:51:04.986863Z","signed_message":"canonical_sha256_bytes"},"source_id":"1902.04101","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2b94cdbb4366fc40c5b16b2d23695dfb5a61e9256660251c11ae453b7f27347d","sha256:a07a380b2527c8a81edbd23c69214aac11b21ec6fc1e7155b44907634dac5700"],"state_sha256":"9334f713868d89a5b7ec880deb55418fa5d63f6733ff0cdd67697213476c5068"}