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We prove that $\\mathcal{K}^*$ is categorical in every $\\xi >\\lambda^+$, $\\mathcal{K}_{\\ge\\beth_{(2^{\\lambda^+})^+}} \\subseteq \\mathcal{K}^{*}$, and the models of $\\mathcal{K}^*_{>\\lambda^+}$ are essentially vector spaces (or trivial i.e. disintegrated)."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1712.08532","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2017-12-22T15:49:18Z","cross_cats_sorted":[],"title_canon_sha256":"c16fdc2e25a63b4a04b22a41f710fa512bee2e06a478e2b930f755f96b677ede","abstract_canon_sha256":"884b2814b567aab73922e33337791ccb727ae00650f4e8446d29903dc9d469bf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:26:23.193267Z","signature_b64":"K+sr7XKfLBVrtrYvTWfOkNFXTv+d9t/kJZrMp1yipYwTtNow6UOwJqPUpilQW8zbW95GyBvhGQdBEsgPkegjCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"959980ccb57d130c25556a75673045e4963e4746ebb451c50653fbc5e2f57ccf","last_reissued_at":"2026-05-18T00:26:23.192628Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:26:23.192628Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Categoricity and Universal Classes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Kaisa Kangas, Tapani Hyttinen","submitted_at":"2017-12-22T15:49:18Z","abstract_excerpt":"Let $(\\mathcal{K} ,\\subseteq )$ be a universal class with $LS(\\mathcal{K})=\\lambda$ categorical in regular $\\kappa >\\lambda^+$ with arbitrarily large models, and let $\\mathcal{K}^*$ be the class of all $\\mathcal{A}\\in\\mathcal{K}_{>\\lambda}$ for which there is $\\mathcal{B} \\in \\mathcal{K}_{\\ge\\kappa}$ such that $\\mathcal{A}\\subseteq\\mathcal{B}$. We prove that $\\mathcal{K}^*$ is categorical in every $\\xi >\\lambda^+$, $\\mathcal{K}_{\\ge\\beth_{(2^{\\lambda^+})^+}} \\subseteq \\mathcal{K}^{*}$, and the models of $\\mathcal{K}^*_{>\\lambda^+}$ are essentially vector spaces (or trivial i.e. disintegrated)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.08532","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1712.08532","created_at":"2026-05-18T00:26:23.192746+00:00"},{"alias_kind":"arxiv_version","alias_value":"1712.08532v2","created_at":"2026-05-18T00:26:23.192746+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.08532","created_at":"2026-05-18T00:26:23.192746+00:00"},{"alias_kind":"pith_short_12","alias_value":"SWMYBTFVPUJQ","created_at":"2026-05-18T12:31:43.269735+00:00"},{"alias_kind":"pith_short_16","alias_value":"SWMYBTFVPUJQYJKV","created_at":"2026-05-18T12:31:43.269735+00:00"},{"alias_kind":"pith_short_8","alias_value":"SWMYBTFV","created_at":"2026-05-18T12:31:43.269735+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/SWMYBTFVPUJQYJKVNJ2WOMCF4S","json":"https://pith.science/pith/SWMYBTFVPUJQYJKVNJ2WOMCF4S.json","graph_json":"https://pith.science/api/pith-number/SWMYBTFVPUJQYJKVNJ2WOMCF4S/graph.json","events_json":"https://pith.science/api/pith-number/SWMYBTFVPUJQYJKVNJ2WOMCF4S/events.json","paper":"https://pith.science/paper/SWMYBTFV"},"agent_actions":{"view_html":"https://pith.science/pith/SWMYBTFVPUJQYJKVNJ2WOMCF4S","download_json":"https://pith.science/pith/SWMYBTFVPUJQYJKVNJ2WOMCF4S.json","view_paper":"https://pith.science/paper/SWMYBTFV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1712.08532&json=true","fetch_graph":"https://pith.science/api/pith-number/SWMYBTFVPUJQYJKVNJ2WOMCF4S/graph.json","fetch_events":"https://pith.science/api/pith-number/SWMYBTFVPUJQYJKVNJ2WOMCF4S/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SWMYBTFVPUJQYJKVNJ2WOMCF4S/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SWMYBTFVPUJQYJKVNJ2WOMCF4S/action/storage_attestation","attest_author":"https://pith.science/pith/SWMYBTFVPUJQYJKVNJ2WOMCF4S/action/author_attestation","sign_citation":"https://pith.science/pith/SWMYBTFVPUJQYJKVNJ2WOMCF4S/action/citation_signature","submit_replication":"https://pith.science/pith/SWMYBTFVPUJQYJKVNJ2WOMCF4S/action/replication_record"}},"created_at":"2026-05-18T00:26:23.192746+00:00","updated_at":"2026-05-18T00:26:23.192746+00:00"}