{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1996:HFQPFKEMJGWWYE4G6R3JMDZVDF","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":"3058565f928fcac85b79d51af38a22fcbb06c12a777bb1f10431d7ef1817a098","cross_cats_sorted":[],"license":"","primary_cat":"math.LO","submitted_at":"1996-08-06T00:00:00Z","title_canon_sha256":"30a21c17cde4430344329a6c7614582e99aa7fcf59be5d1152d11601bae912f8"},"schema_version":"1.0","source":{"id":"math/9608203","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9608203","created_at":"2026-05-18T01:05:47Z"},{"alias_kind":"arxiv_version","alias_value":"math/9608203v1","created_at":"2026-05-18T01:05:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9608203","created_at":"2026-05-18T01:05:47Z"},{"alias_kind":"pith_short_12","alias_value":"HFQPFKEMJGWW","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"HFQPFKEMJGWWYE4G","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"HFQPFKEM","created_at":"2026-05-18T12:25:47Z"}],"graph_snapshots":[{"event_id":"sha256:216dbe77cc99e77feb0decf3d8d7003c58eaaf9605b7b39370e2afc24b7af7df","target":"graph","created_at":"2026-05-18T01:05:47Z","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":"Let D = { d_n } be a countable collection of Delta^1_3 degrees. Assuming that all co-analytic games on integers are determined (or equivalently that all reals have ``sharps''), we prove that either D has  a Delta^1_3-minimal upper bound, or that for any n, and for every real r recursive in d_n, games in the pointclasses Delta^1_2(r) are determined. This is proven using Core Model theory.","authors_text":"Philip Welch","cross_cats":[],"headline":"","license":"","primary_cat":"math.LO","submitted_at":"1996-08-06T00:00:00Z","title":"Determinacy and \\Delta^1_3-degrees"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9608203","kind":"arxiv","version":1},"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":"sha256:2839edea8bdf9d9d6b21b95be51faf88e7446feaf88100a5bea9dc037ae8bd31","target":"record","created_at":"2026-05-18T01:05:47Z","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":"3058565f928fcac85b79d51af38a22fcbb06c12a777bb1f10431d7ef1817a098","cross_cats_sorted":[],"license":"","primary_cat":"math.LO","submitted_at":"1996-08-06T00:00:00Z","title_canon_sha256":"30a21c17cde4430344329a6c7614582e99aa7fcf59be5d1152d11601bae912f8"},"schema_version":"1.0","source":{"id":"math/9608203","kind":"arxiv","version":1}},"canonical_sha256":"3960f2a88c49ad6c1386f476960f351943844734d87261012f087d3412fde827","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3960f2a88c49ad6c1386f476960f351943844734d87261012f087d3412fde827","first_computed_at":"2026-05-18T01:05:47.134378Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:47.134378Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TstfloKm9KaYIot7Lw7vDj7vsttedU0TleH+ioAf4YhmZNXhTjZGwjE3bP/zpFUyM66VUg3X9ZPTgHGUmqLaDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:47.134923Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/9608203","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2839edea8bdf9d9d6b21b95be51faf88e7446feaf88100a5bea9dc037ae8bd31","sha256:216dbe77cc99e77feb0decf3d8d7003c58eaaf9605b7b39370e2afc24b7af7df"],"state_sha256":"2694460b57d330238ee776706f2d1ca78ed597ed90f008b1267cc927dbf215c2"}