{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:7SRJ5ZFGCQWD5Z5XDINE5M4DJT","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":"f9346298b70bab82c125091cf6a88bb54bb2fce5ef3dd6ed5974b1ec54b516ac","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2017-12-03T13:47:31Z","title_canon_sha256":"b8071de90124ace920c53a6d54d7b72fced6ad4a2f60fd14a1a47cf104a88b38"},"schema_version":"1.0","source":{"id":"1712.00769","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1712.00769","created_at":"2026-05-18T00:01:30Z"},{"alias_kind":"arxiv_version","alias_value":"1712.00769v3","created_at":"2026-05-18T00:01:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.00769","created_at":"2026-05-18T00:01:30Z"},{"alias_kind":"pith_short_12","alias_value":"7SRJ5ZFGCQWD","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_16","alias_value":"7SRJ5ZFGCQWD5Z5X","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_8","alias_value":"7SRJ5ZFG","created_at":"2026-05-18T12:31:05Z"}],"graph_snapshots":[{"event_id":"sha256:a0e0f760ac22d12ca3433295dfdfbffe1ffbfb732b437fc98606930bba45acdc","target":"graph","created_at":"2026-05-18T00:01:30Z","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":"We present a model of set theory, in which, for a given $n\\ge2$, there exists a non-ROD-uniformizable planar lightface $\\varPi^1_n$ set in $\\mathbb R\\times\\mathbb R$, whose all vertical cross-sections are countable sets (and in fact Vitali classes), while all planar boldface $\\bf\\Sigma^1_n$ sets with countable cross-sections are $\\bf\\Delta^1_{n+1}$-uniformizable. Thus it is true in this model, that the ROD-uniformization principle for sets with countable cross-sections first fails precisely at a given projective level.","authors_text":"Vassily Lyubetsky, Vladimir Kanovei","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2017-12-03T13:47:31Z","title":"Non-uniformizable sets with countable cross-sections on a given level of the projective hierarchy"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.00769","kind":"arxiv","version":3},"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":"sha256:40313f4cce9b2b889121ee330c531819585c76c39923ef2029d68529482b713e","target":"record","created_at":"2026-05-18T00:01:30Z","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":"f9346298b70bab82c125091cf6a88bb54bb2fce5ef3dd6ed5974b1ec54b516ac","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2017-12-03T13:47:31Z","title_canon_sha256":"b8071de90124ace920c53a6d54d7b72fced6ad4a2f60fd14a1a47cf104a88b38"},"schema_version":"1.0","source":{"id":"1712.00769","kind":"arxiv","version":3}},"canonical_sha256":"fca29ee4a6142c3ee7b71a1a4eb3834ce04518ac37c969ccb84cd184fcc6fed8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fca29ee4a6142c3ee7b71a1a4eb3834ce04518ac37c969ccb84cd184fcc6fed8","first_computed_at":"2026-05-18T00:01:30.331089Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:01:30.331089Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"iZ4oBzmm5jPVnr0ZxoFEYXVmWyOEzB/zD3nibMDxRal3vBRNu2DoNbmxy5d5N/CnUu3CoNP58biRG3iDDVJEDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:01:30.331551Z","signed_message":"canonical_sha256_bytes"},"source_id":"1712.00769","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:40313f4cce9b2b889121ee330c531819585c76c39923ef2029d68529482b713e","sha256:a0e0f760ac22d12ca3433295dfdfbffe1ffbfb732b437fc98606930bba45acdc"],"state_sha256":"0c7bbdd6611c5ae28421c28662296fcfa74331153e41a0b169e79afc93d6dce0"}