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If $\\boldsymbol\\Pi^1_n$ determinacy and $\\Pi^1_{n+1}$ determinacy both hold true and there is no $\\boldsymbol\\Sigma^1_{n+2}$-definable $\\omega_1$-sequence of pairwise distinct reals, then $M_n^\\#$ exists and is $\\omega_1$-iterable. The proof yields that $\\boldsymbol\\Pi^1_{n+1}$ determinacy implies that $M_n^\\#(x)$ exists and is $\\omega_1$-iterable for all reals $x$. A consequence is the Determinacy Transfer Theorem for arbitrary $n \\geq 1$, namely the statement that $\\boldsymbol\\Pi^1_{n+1}$ determinacy implies $\\Ga"},"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":"1902.05890","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2019-02-15T17:16:03Z","cross_cats_sorted":[],"title_canon_sha256":"e76c9bcd719cf6775910a8d1fb513778f88eb225fa6a1323877cb7e4db47c73d","abstract_canon_sha256":"6e8aff3d719a8a3d9919bb418446c4eec48c6d8b1b18e3ca9669c7141a23a451"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:53:55.193590Z","signature_b64":"9LHt+Pzcv9ECFi5lQim5FFPQcA1gERdxxIhKs45sVskli/wmiClK4QAbcw9sHbXG+h8aWnw7fWMYlGfEwdreBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"724697aea1062b6cfe93ae8a9f56503bf19d081f437dd202579b14a3be280bc1","last_reissued_at":"2026-05-17T23:53:55.192856Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:53:55.192856Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Mice with finitely many Woodin cardinals from optimal determinacy hypotheses","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Ralf Schindler, Sandra M\\\"uller, W. Hugh Woodin","submitted_at":"2019-02-15T17:16:03Z","abstract_excerpt":"We prove the following result which is due to the third author. Let $n \\geq 1$. If $\\boldsymbol\\Pi^1_n$ determinacy and $\\Pi^1_{n+1}$ determinacy both hold true and there is no $\\boldsymbol\\Sigma^1_{n+2}$-definable $\\omega_1$-sequence of pairwise distinct reals, then $M_n^\\#$ exists and is $\\omega_1$-iterable. The proof yields that $\\boldsymbol\\Pi^1_{n+1}$ determinacy implies that $M_n^\\#(x)$ exists and is $\\omega_1$-iterable for all reals $x$. A consequence is the Determinacy Transfer Theorem for arbitrary $n \\geq 1$, namely the statement that $\\boldsymbol\\Pi^1_{n+1}$ determinacy implies $\\Ga"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.05890","kind":"arxiv","version":1},"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":"1902.05890","created_at":"2026-05-17T23:53:55.192970+00:00"},{"alias_kind":"arxiv_version","alias_value":"1902.05890v1","created_at":"2026-05-17T23:53:55.192970+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1902.05890","created_at":"2026-05-17T23:53:55.192970+00:00"},{"alias_kind":"pith_short_12","alias_value":"OJDJPLVBAYVW","created_at":"2026-05-18T12:33:24.271573+00:00"},{"alias_kind":"pith_short_16","alias_value":"OJDJPLVBAYVWZ7UT","created_at":"2026-05-18T12:33:24.271573+00:00"},{"alias_kind":"pith_short_8","alias_value":"OJDJPLVB","created_at":"2026-05-18T12:33:24.271573+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/OJDJPLVBAYVWZ7UTV2FJ6VSQHP","json":"https://pith.science/pith/OJDJPLVBAYVWZ7UTV2FJ6VSQHP.json","graph_json":"https://pith.science/api/pith-number/OJDJPLVBAYVWZ7UTV2FJ6VSQHP/graph.json","events_json":"https://pith.science/api/pith-number/OJDJPLVBAYVWZ7UTV2FJ6VSQHP/events.json","paper":"https://pith.science/paper/OJDJPLVB"},"agent_actions":{"view_html":"https://pith.science/pith/OJDJPLVBAYVWZ7UTV2FJ6VSQHP","download_json":"https://pith.science/pith/OJDJPLVBAYVWZ7UTV2FJ6VSQHP.json","view_paper":"https://pith.science/paper/OJDJPLVB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1902.05890&json=true","fetch_graph":"https://pith.science/api/pith-number/OJDJPLVBAYVWZ7UTV2FJ6VSQHP/graph.json","fetch_events":"https://pith.science/api/pith-number/OJDJPLVBAYVWZ7UTV2FJ6VSQHP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OJDJPLVBAYVWZ7UTV2FJ6VSQHP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OJDJPLVBAYVWZ7UTV2FJ6VSQHP/action/storage_attestation","attest_author":"https://pith.science/pith/OJDJPLVBAYVWZ7UTV2FJ6VSQHP/action/author_attestation","sign_citation":"https://pith.science/pith/OJDJPLVBAYVWZ7UTV2FJ6VSQHP/action/citation_signature","submit_replication":"https://pith.science/pith/OJDJPLVBAYVWZ7UTV2FJ6VSQHP/action/replication_record"}},"created_at":"2026-05-17T23:53:55.192970+00:00","updated_at":"2026-05-17T23:53:55.192970+00:00"}