{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:JI4UBI3HO2TH55TMEGBJHCZ4ED","short_pith_number":"pith:JI4UBI3H","schema_version":"1.0","canonical_sha256":"4a3940a36776a67ef66c2182938b3c20ce68055e7e77ee0504815b94b8fbacee","source":{"kind":"arxiv","id":"1011.0365","version":1},"attestation_state":"computed","paper":{"title":"Bi-Lipschitz Embeddability of the Grushin Plane into Euclidean Space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Jeehyeon Seo","submitted_at":"2010-11-01T16:19:56Z","abstract_excerpt":"Many sub-Riemannian manifolds like the Heisenberg group do not admit bi- Lipschitz embedding into any Euclidean space. In contrast, the Grushin plane admits a bi-Lipschitz embedding into some Euclidean space. This is done by extending a bi-Lipschitz embedding of the singular line, using a Whitney decomposition of its complement."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1011.0365","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2010-11-01T16:19:56Z","cross_cats_sorted":[],"title_canon_sha256":"a2d1d444771f136f7b3ce7464f77bf79b30e813edf7e7701bf50c06a797a47de","abstract_canon_sha256":"ee2eee66e3fcf2b439c30578578620e80f38e302ea189eb169483d08f13fc340"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:37:23.335665Z","signature_b64":"F84X8ToPAd+I+P0RTJaPF0YHSt8PaefoEoG1TUodmBOUVFWvr8lnPgjGCXD5Ru56mraFFPH4Pc4fVINKdC7mBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4a3940a36776a67ef66c2182938b3c20ce68055e7e77ee0504815b94b8fbacee","last_reissued_at":"2026-05-18T04:37:23.335219Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:37:23.335219Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bi-Lipschitz Embeddability of the Grushin Plane into Euclidean Space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Jeehyeon Seo","submitted_at":"2010-11-01T16:19:56Z","abstract_excerpt":"Many sub-Riemannian manifolds like the Heisenberg group do not admit bi- Lipschitz embedding into any Euclidean space. In contrast, the Grushin plane admits a bi-Lipschitz embedding into some Euclidean space. This is done by extending a bi-Lipschitz embedding of the singular line, using a Whitney decomposition of its complement."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.0365","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1011.0365","created_at":"2026-05-18T04:37:23.335277+00:00"},{"alias_kind":"arxiv_version","alias_value":"1011.0365v1","created_at":"2026-05-18T04:37:23.335277+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.0365","created_at":"2026-05-18T04:37:23.335277+00:00"},{"alias_kind":"pith_short_12","alias_value":"JI4UBI3HO2TH","created_at":"2026-05-18T12:26:09.077623+00:00"},{"alias_kind":"pith_short_16","alias_value":"JI4UBI3HO2TH55TM","created_at":"2026-05-18T12:26:09.077623+00:00"},{"alias_kind":"pith_short_8","alias_value":"JI4UBI3H","created_at":"2026-05-18T12:26:09.077623+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/JI4UBI3HO2TH55TMEGBJHCZ4ED","json":"https://pith.science/pith/JI4UBI3HO2TH55TMEGBJHCZ4ED.json","graph_json":"https://pith.science/api/pith-number/JI4UBI3HO2TH55TMEGBJHCZ4ED/graph.json","events_json":"https://pith.science/api/pith-number/JI4UBI3HO2TH55TMEGBJHCZ4ED/events.json","paper":"https://pith.science/paper/JI4UBI3H"},"agent_actions":{"view_html":"https://pith.science/pith/JI4UBI3HO2TH55TMEGBJHCZ4ED","download_json":"https://pith.science/pith/JI4UBI3HO2TH55TMEGBJHCZ4ED.json","view_paper":"https://pith.science/paper/JI4UBI3H","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1011.0365&json=true","fetch_graph":"https://pith.science/api/pith-number/JI4UBI3HO2TH55TMEGBJHCZ4ED/graph.json","fetch_events":"https://pith.science/api/pith-number/JI4UBI3HO2TH55TMEGBJHCZ4ED/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JI4UBI3HO2TH55TMEGBJHCZ4ED/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JI4UBI3HO2TH55TMEGBJHCZ4ED/action/storage_attestation","attest_author":"https://pith.science/pith/JI4UBI3HO2TH55TMEGBJHCZ4ED/action/author_attestation","sign_citation":"https://pith.science/pith/JI4UBI3HO2TH55TMEGBJHCZ4ED/action/citation_signature","submit_replication":"https://pith.science/pith/JI4UBI3HO2TH55TMEGBJHCZ4ED/action/replication_record"}},"created_at":"2026-05-18T04:37:23.335277+00:00","updated_at":"2026-05-18T04:37:23.335277+00:00"}