{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:OCUF3DY774QFEW3T7E5MBAJYV3","short_pith_number":"pith:OCUF3DY7","canonical_record":{"source":{"id":"1504.02083","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2015-04-08T19:22:59Z","cross_cats_sorted":["math.SG"],"title_canon_sha256":"456b46f9507334b5d27faccf6b0ece0ff67304dfab3be7f32d8f5559529dcf1b","abstract_canon_sha256":"d509e7af5ee6549674168efa3170b8d859fd94ad604d898cceb506409ac3ac46"},"schema_version":"1.0"},"canonical_sha256":"70a85d8f1fff20525b73f93ac08138aec75e7bc75335c4bac3c048ba9a4a7d52","source":{"kind":"arxiv","id":"1504.02083","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.02083","created_at":"2026-05-18T00:58:35Z"},{"alias_kind":"arxiv_version","alias_value":"1504.02083v2","created_at":"2026-05-18T00:58:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.02083","created_at":"2026-05-18T00:58:35Z"},{"alias_kind":"pith_short_12","alias_value":"OCUF3DY774QF","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_16","alias_value":"OCUF3DY774QFEW3T","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_8","alias_value":"OCUF3DY7","created_at":"2026-05-18T12:29:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:OCUF3DY774QFEW3T7E5MBAJYV3","target":"record","payload":{"canonical_record":{"source":{"id":"1504.02083","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2015-04-08T19:22:59Z","cross_cats_sorted":["math.SG"],"title_canon_sha256":"456b46f9507334b5d27faccf6b0ece0ff67304dfab3be7f32d8f5559529dcf1b","abstract_canon_sha256":"d509e7af5ee6549674168efa3170b8d859fd94ad604d898cceb506409ac3ac46"},"schema_version":"1.0"},"canonical_sha256":"70a85d8f1fff20525b73f93ac08138aec75e7bc75335c4bac3c048ba9a4a7d52","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:58:35.841903Z","signature_b64":"f6x53EglRLVrvJrINZdAMQRfCHTcYLolo9IsRIhnnam1bkH3APwg86CdCdQZTm1lKUIUdtmo2PlGswaIkf6MBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"70a85d8f1fff20525b73f93ac08138aec75e7bc75335c4bac3c048ba9a4a7d52","last_reissued_at":"2026-05-18T00:58:35.841385Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:58:35.841385Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1504.02083","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:58:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sD1UHnu6gXaJbZ43JCQI3YmJUSrZMn4g2ekFEci2BmRdFkFrDUk4l80bkwGF9IcJQGe+1Dp30yX96Yj1GNLNAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T19:48:47.366966Z"},"content_sha256":"8af805e3957d24c895c8a8eb35690157433f20f93145aeca7522ff296ecb4266","schema_version":"1.0","event_id":"sha256:8af805e3957d24c895c8a8eb35690157433f20f93145aeca7522ff296ecb4266"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:OCUF3DY774QFEW3T7E5MBAJYV3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Rational approximation and Lagrangian inclusions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SG"],"primary_cat":"math.CV","authors_text":"Alexandre Sukhov, Rasul Shafikov","submitted_at":"2015-04-08T19:22:59Z","abstract_excerpt":"We show that a Lagrangian inclusion in $\\mathbb C^2$ with double transverse self-intersection points and standard open Whitney umbrellas is rationally convex. As an application we show that any compact surface $S$, except $S^2$ and $\\mathbb RP_2$, admits a pair of smooth complex-valued functions $f_1$, $f_2$ with the property that any continuous complex valued function on $S$ is a uniform limit of a sequence of $R_j(f_1,f_2)$, where $R_j(z_1,z_2)$ are rational functions in $\\mathbb C^2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.02083","kind":"arxiv","version":2},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:58:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KN7JdF3Lxcbi3a6v1yYwi4Z0fYwOS4IxmuBGkvEfREKytU1Z3UuJJvLLMo5HzXcbsB7z00zwHFp2Ae5Iw/8jDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T19:48:47.367631Z"},"content_sha256":"8b4f29a3cfe693dc70b98ab309f9c37a37c7d6c1c55fc435cfbde692ecd4d97d","schema_version":"1.0","event_id":"sha256:8b4f29a3cfe693dc70b98ab309f9c37a37c7d6c1c55fc435cfbde692ecd4d97d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OCUF3DY774QFEW3T7E5MBAJYV3/bundle.json","state_url":"https://pith.science/pith/OCUF3DY774QFEW3T7E5MBAJYV3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OCUF3DY774QFEW3T7E5MBAJYV3/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-30T19:48:47Z","links":{"resolver":"https://pith.science/pith/OCUF3DY774QFEW3T7E5MBAJYV3","bundle":"https://pith.science/pith/OCUF3DY774QFEW3T7E5MBAJYV3/bundle.json","state":"https://pith.science/pith/OCUF3DY774QFEW3T7E5MBAJYV3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OCUF3DY774QFEW3T7E5MBAJYV3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:OCUF3DY774QFEW3T7E5MBAJYV3","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":"d509e7af5ee6549674168efa3170b8d859fd94ad604d898cceb506409ac3ac46","cross_cats_sorted":["math.SG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2015-04-08T19:22:59Z","title_canon_sha256":"456b46f9507334b5d27faccf6b0ece0ff67304dfab3be7f32d8f5559529dcf1b"},"schema_version":"1.0","source":{"id":"1504.02083","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.02083","created_at":"2026-05-18T00:58:35Z"},{"alias_kind":"arxiv_version","alias_value":"1504.02083v2","created_at":"2026-05-18T00:58:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.02083","created_at":"2026-05-18T00:58:35Z"},{"alias_kind":"pith_short_12","alias_value":"OCUF3DY774QF","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_16","alias_value":"OCUF3DY774QFEW3T","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_8","alias_value":"OCUF3DY7","created_at":"2026-05-18T12:29:34Z"}],"graph_snapshots":[{"event_id":"sha256:8b4f29a3cfe693dc70b98ab309f9c37a37c7d6c1c55fc435cfbde692ecd4d97d","target":"graph","created_at":"2026-05-18T00:58:35Z","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 show that a Lagrangian inclusion in $\\mathbb C^2$ with double transverse self-intersection points and standard open Whitney umbrellas is rationally convex. As an application we show that any compact surface $S$, except $S^2$ and $\\mathbb RP_2$, admits a pair of smooth complex-valued functions $f_1$, $f_2$ with the property that any continuous complex valued function on $S$ is a uniform limit of a sequence of $R_j(f_1,f_2)$, where $R_j(z_1,z_2)$ are rational functions in $\\mathbb C^2$.","authors_text":"Alexandre Sukhov, Rasul Shafikov","cross_cats":["math.SG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2015-04-08T19:22:59Z","title":"Rational approximation and Lagrangian inclusions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.02083","kind":"arxiv","version":2},"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":"sha256:8af805e3957d24c895c8a8eb35690157433f20f93145aeca7522ff296ecb4266","target":"record","created_at":"2026-05-18T00:58:35Z","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":"d509e7af5ee6549674168efa3170b8d859fd94ad604d898cceb506409ac3ac46","cross_cats_sorted":["math.SG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2015-04-08T19:22:59Z","title_canon_sha256":"456b46f9507334b5d27faccf6b0ece0ff67304dfab3be7f32d8f5559529dcf1b"},"schema_version":"1.0","source":{"id":"1504.02083","kind":"arxiv","version":2}},"canonical_sha256":"70a85d8f1fff20525b73f93ac08138aec75e7bc75335c4bac3c048ba9a4a7d52","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"70a85d8f1fff20525b73f93ac08138aec75e7bc75335c4bac3c048ba9a4a7d52","first_computed_at":"2026-05-18T00:58:35.841385Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:58:35.841385Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"f6x53EglRLVrvJrINZdAMQRfCHTcYLolo9IsRIhnnam1bkH3APwg86CdCdQZTm1lKUIUdtmo2PlGswaIkf6MBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:58:35.841903Z","signed_message":"canonical_sha256_bytes"},"source_id":"1504.02083","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8af805e3957d24c895c8a8eb35690157433f20f93145aeca7522ff296ecb4266","sha256:8b4f29a3cfe693dc70b98ab309f9c37a37c7d6c1c55fc435cfbde692ecd4d97d"],"state_sha256":"fcbdc6c48e8864684a425a9ebe963a51a7e3eaf9d633f92c43afb30d3872913f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YIc1Ae4XBdqtVNuDAJi7VJsf0kBixyIQ1zL6CEf13jTIwJKLEeooT69bgF6kaSk2HNRon6r4AISlCz3b5pX4BQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T19:48:47.371128Z","bundle_sha256":"1d9a925788620dd7cdd125694c4eaa7322cbd5fcd558490774902d6385132a34"}}