{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:6QDLT6WEHWCIUC6ZOYBY6LYG25","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":"e3492980132283a9c94536c77ce0f54344ac18284f5a90e8aaa8067170b4e59c","cross_cats_sorted":["math.LO"],"license":"","primary_cat":"cs.LO","submitted_at":"2006-10-13T07:45:05Z","title_canon_sha256":"60f8c75ba46be470f7bf16338148ee0480725167632fc1230caf1bbe5ff8ca22"},"schema_version":"1.0","source":{"id":"cs/0610080","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"cs/0610080","created_at":"2026-05-18T04:16:21Z"},{"alias_kind":"arxiv_version","alias_value":"cs/0610080v5","created_at":"2026-05-18T04:16:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.cs/0610080","created_at":"2026-05-18T04:16:21Z"},{"alias_kind":"pith_short_12","alias_value":"6QDLT6WEHWCI","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"6QDLT6WEHWCIUC6Z","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"6QDLT6WE","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:24568cf0653d93cb962446cd7023e070fba8d7d060c751516bc9e4202d4d1998","target":"graph","created_at":"2026-05-18T04:16:21Z","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":"The empty set of course contains no computable point. On the other hand, surprising results due to Zaslavskii, Tseitin, Kreisel, and Lacombe assert the existence of NON-empty co-r.e. closed sets devoid of computable points: sets which are `large' in the sense of positive Lebesgue measure. We observe that a certain size is in fact necessary: every non-empty co-r.e. closed real set without computable points has continuum cardinality.\n  This leads us to investigate for various classes of computable real subsets whether they necessarily contain a (not necessarily effectively findable) computable p","authors_text":"Martin Ziegler, St\\'ephane Le Roux","cross_cats":["math.LO"],"headline":"","license":"","primary_cat":"cs.LO","submitted_at":"2006-10-13T07:45:05Z","title":"Computable Closed Euclidean Subsets with and without Computable Points"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"cs/0610080","kind":"arxiv","version":5},"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":"sha256:7718dd773bfa2b0d3c75c1dd490e910a04d8040ccd38ed40a427960e99ffe01e","target":"record","created_at":"2026-05-18T04:16:21Z","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":"e3492980132283a9c94536c77ce0f54344ac18284f5a90e8aaa8067170b4e59c","cross_cats_sorted":["math.LO"],"license":"","primary_cat":"cs.LO","submitted_at":"2006-10-13T07:45:05Z","title_canon_sha256":"60f8c75ba46be470f7bf16338148ee0480725167632fc1230caf1bbe5ff8ca22"},"schema_version":"1.0","source":{"id":"cs/0610080","kind":"arxiv","version":5}},"canonical_sha256":"f406b9fac43d848a0bd976038f2f06d77de0be98ad76bb087a3d078bbba26670","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f406b9fac43d848a0bd976038f2f06d77de0be98ad76bb087a3d078bbba26670","first_computed_at":"2026-05-18T04:16:21.750037Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:16:21.750037Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Y5YnYrCAyRirK6d2sKoQj8Lvse/VzOgGymqpPg5LFEP51oGCdnfIh1XFdV5bbU184YgUmTFFiKa7adiqvLpyDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:16:21.750659Z","signed_message":"canonical_sha256_bytes"},"source_id":"cs/0610080","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7718dd773bfa2b0d3c75c1dd490e910a04d8040ccd38ed40a427960e99ffe01e","sha256:24568cf0653d93cb962446cd7023e070fba8d7d060c751516bc9e4202d4d1998"],"state_sha256":"66fa40709f1f5094501403a6455821eae068eddaf7b36474cfebed9ae973a048"}