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This improves several classical results of Shelah.\n  $\\mathbf{Theorem}$\n  Let $\\mu \\ge \\text{LS} (K)$. If $K$ is categorical in a $\\lambda \\ge \\beth_{\\left(2^{\\mu}\\right)^+}$, then:\n  1) Whenever $M_0, M_1, M_2 \\in K_\\mu$ are such that $M_1$ and $M_2$ are limit over $M_0$, we have $M_1 \\cong_{M_0} M_2$.\n  2) If $\\mu > \\text{LS} (K)$, the model of size $\\lambda$ is "},"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":"1509.01488","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-09-04T15:14:34Z","cross_cats_sorted":[],"title_canon_sha256":"cd68647c3afcbff3663ba83066ab0ec15da24bdb7e3cbebc997f129cd62024c6","abstract_canon_sha256":"3ad3e0df35fea9549a748dcc3964a8f561675a1760dd98ab7e2a1d82efe7b5cb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:20:29.352272Z","signature_b64":"e+8bLtBR6moPCGi+jJhzPOY6DHyI4dysLxal5vC7OMl0KpThto8gkBzvwdGbluS0N+3NhvSp89N9XqhBK1axBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1e6390fca794856e74f75f8907c140a9ce401d5f2cdf54c810ff3cfada63fa9f","last_reissued_at":"2026-05-18T01:20:29.351556Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:20:29.351556Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the structure of categorical abstract elementary classes with amalgamation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Monica M. VanDieren, Sebastien Vasey","submitted_at":"2015-09-04T15:14:34Z","abstract_excerpt":"For $K$ an abstract elementary class with amalgamation and no maximal models, we show that categoricity in a high-enough cardinal implies structural properties such as the uniqueness of limit models and the existence of good frames. This improves several classical results of Shelah.\n  $\\mathbf{Theorem}$\n  Let $\\mu \\ge \\text{LS} (K)$. If $K$ is categorical in a $\\lambda \\ge \\beth_{\\left(2^{\\mu}\\right)^+}$, then:\n  1) Whenever $M_0, M_1, M_2 \\in K_\\mu$ are such that $M_1$ and $M_2$ are limit over $M_0$, we have $M_1 \\cong_{M_0} M_2$.\n  2) If $\\mu > \\text{LS} (K)$, the model of size $\\lambda$ is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.01488","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1509.01488","created_at":"2026-05-18T01:20:29.351679+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.01488v3","created_at":"2026-05-18T01:20:29.351679+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.01488","created_at":"2026-05-18T01:20:29.351679+00:00"},{"alias_kind":"pith_short_12","alias_value":"DZRZB7FHSSCW","created_at":"2026-05-18T12:29:17.054201+00:00"},{"alias_kind":"pith_short_16","alias_value":"DZRZB7FHSSCW45HX","created_at":"2026-05-18T12:29:17.054201+00:00"},{"alias_kind":"pith_short_8","alias_value":"DZRZB7FH","created_at":"2026-05-18T12:29:17.054201+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/DZRZB7FHSSCW45HXL6EQPQKAVH","json":"https://pith.science/pith/DZRZB7FHSSCW45HXL6EQPQKAVH.json","graph_json":"https://pith.science/api/pith-number/DZRZB7FHSSCW45HXL6EQPQKAVH/graph.json","events_json":"https://pith.science/api/pith-number/DZRZB7FHSSCW45HXL6EQPQKAVH/events.json","paper":"https://pith.science/paper/DZRZB7FH"},"agent_actions":{"view_html":"https://pith.science/pith/DZRZB7FHSSCW45HXL6EQPQKAVH","download_json":"https://pith.science/pith/DZRZB7FHSSCW45HXL6EQPQKAVH.json","view_paper":"https://pith.science/paper/DZRZB7FH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.01488&json=true","fetch_graph":"https://pith.science/api/pith-number/DZRZB7FHSSCW45HXL6EQPQKAVH/graph.json","fetch_events":"https://pith.science/api/pith-number/DZRZB7FHSSCW45HXL6EQPQKAVH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DZRZB7FHSSCW45HXL6EQPQKAVH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DZRZB7FHSSCW45HXL6EQPQKAVH/action/storage_attestation","attest_author":"https://pith.science/pith/DZRZB7FHSSCW45HXL6EQPQKAVH/action/author_attestation","sign_citation":"https://pith.science/pith/DZRZB7FHSSCW45HXL6EQPQKAVH/action/citation_signature","submit_replication":"https://pith.science/pith/DZRZB7FHSSCW45HXL6EQPQKAVH/action/replication_record"}},"created_at":"2026-05-18T01:20:29.351679+00:00","updated_at":"2026-05-18T01:20:29.351679+00:00"}