{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2002:ZOK5LLBXMNBMFOREOW3LLSNY3F","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":"4307792ddccb4340e98aaf8809326e65d9fe5c86f9e47da36c27cd4c74ff0a12","cross_cats_sorted":[],"license":"","primary_cat":"math.GT","submitted_at":"2002-05-04T09:24:47Z","title_canon_sha256":"131cd450b59f35988d0f2a7d96e96166113f62bdbe246cb459644525ea012fbc"},"schema_version":"1.0","source":{"id":"math/0205036","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0205036","created_at":"2026-05-18T02:41:33Z"},{"alias_kind":"arxiv_version","alias_value":"math/0205036v1","created_at":"2026-05-18T02:41:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0205036","created_at":"2026-05-18T02:41:33Z"},{"alias_kind":"pith_short_12","alias_value":"ZOK5LLBXMNBM","created_at":"2026-05-18T12:25:51Z"},{"alias_kind":"pith_short_16","alias_value":"ZOK5LLBXMNBMFORE","created_at":"2026-05-18T12:25:51Z"},{"alias_kind":"pith_short_8","alias_value":"ZOK5LLBX","created_at":"2026-05-18T12:25:51Z"}],"graph_snapshots":[{"event_id":"sha256:ab9e6754ebd38e61886a3d1e77002fcdbb3b7f988a50ad2f2d92623436d7d220","target":"graph","created_at":"2026-05-18T02:41:33Z","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 F be a foliation of codimension 2 on a compact manifold with at least one non-compact leaf. We show that then F must contain uncountably many non-compact leaves. We prove the same statement for oriented p-dimensional foliations of arbitrary codimension if there exists a closed p form which evaluates positively on every compact leaf. For foliations of codimension 1 on compact manifolds it is known that the union of all non-compact leaves is an open set [A Haefliger, Varietes feuilletes, Ann. Scuola Norm. Sup. Pisa 16 (1962) 367-397].","authors_text":"Elmar Vogt","cross_cats":[],"headline":"","license":"","primary_cat":"math.GT","submitted_at":"2002-05-04T09:24:47Z","title":"Foliations with few non-compact leaves"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0205036","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":"sha256:2269b86e27b485be51ef8d8a32067c4462ff79a056ec45caf6a0e7118e5195a2","target":"record","created_at":"2026-05-18T02:41:33Z","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":"4307792ddccb4340e98aaf8809326e65d9fe5c86f9e47da36c27cd4c74ff0a12","cross_cats_sorted":[],"license":"","primary_cat":"math.GT","submitted_at":"2002-05-04T09:24:47Z","title_canon_sha256":"131cd450b59f35988d0f2a7d96e96166113f62bdbe246cb459644525ea012fbc"},"schema_version":"1.0","source":{"id":"math/0205036","kind":"arxiv","version":1}},"canonical_sha256":"cb95d5ac376342c2ba2475b6b5c9b8d94cc175bbc00f3addf1d8774fa4a0998d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cb95d5ac376342c2ba2475b6b5c9b8d94cc175bbc00f3addf1d8774fa4a0998d","first_computed_at":"2026-05-18T02:41:33.056113Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:41:33.056113Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TjVcGhbcapqEGzw0dqsgUuJQoBnQEFa/qcWN8pkcQN18azafUCqKWEJF4jXYAfEUgdTVsmzS0E3ivws6tzZVAg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:41:33.056600Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0205036","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2269b86e27b485be51ef8d8a32067c4462ff79a056ec45caf6a0e7118e5195a2","sha256:ab9e6754ebd38e61886a3d1e77002fcdbb3b7f988a50ad2f2d92623436d7d220"],"state_sha256":"9b4d98005043826f86a051950f4a1cd040b7257490972514f48f841e555565f3"}