{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:QWOZGRCQDS2VITV5WPKOPRFPE4","short_pith_number":"pith:QWOZGRCQ","schema_version":"1.0","canonical_sha256":"859d9344501cb5544ebdb3d4e7c4af272b7f7e640ee9292a864c90c14341d2de","source":{"kind":"arxiv","id":"1703.05421","version":1},"attestation_state":"computed","paper":{"title":"Tangent cones and $C^1$ regularity of definable sets","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.GT","authors_text":"Krzysztof Kurdyka, Nhan Nguyen, Olivier Le Gal","submitted_at":"2017-03-15T23:16:58Z","abstract_excerpt":"Let $X\\subset \\mathbb R^n$ be a connected locally closed definable set in an o-minimal structure. We prove that the following three statements are equivalent: (i) $X$ is a $C^1$ manifold, (ii) the tangent cone and the paratangent cone of $X$ coincide at every point in $X$, (iii) for every $x \\in X$, the tangent cone of $X$ at the point $x$ is a $k$-dimensional linear subspace of $\\mathbb R^n$ ($k$ does not depend on $x$) varies continuously in $x$, and the density $\\theta(X, x) < 3/2$."},"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":"1703.05421","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.GT","submitted_at":"2017-03-15T23:16:58Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"db46f3cc965de3fcba19360e1f6883c44cb26ad21cffb8b36052cd989aa6a0b6","abstract_canon_sha256":"60ce61fb22c457bf39c3dcaeef12663b5e62134b9e384998de0fbca940db4dda"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:48:34.331779Z","signature_b64":"YV5KMap4BrB96/pI+u1vhN0R0Dw9LKGHPZERv7uoxAsJVQ7TVizvK2Lv54AL9GGBgH1kT3JqkrWtlX493SNtCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"859d9344501cb5544ebdb3d4e7c4af272b7f7e640ee9292a864c90c14341d2de","last_reissued_at":"2026-05-18T00:48:34.331248Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:48:34.331248Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Tangent cones and $C^1$ regularity of definable sets","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.GT","authors_text":"Krzysztof Kurdyka, Nhan Nguyen, Olivier Le Gal","submitted_at":"2017-03-15T23:16:58Z","abstract_excerpt":"Let $X\\subset \\mathbb R^n$ be a connected locally closed definable set in an o-minimal structure. We prove that the following three statements are equivalent: (i) $X$ is a $C^1$ manifold, (ii) the tangent cone and the paratangent cone of $X$ coincide at every point in $X$, (iii) for every $x \\in X$, the tangent cone of $X$ at the point $x$ is a $k$-dimensional linear subspace of $\\mathbb R^n$ ($k$ does not depend on $x$) varies continuously in $x$, and the density $\\theta(X, x) < 3/2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.05421","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":"1703.05421","created_at":"2026-05-18T00:48:34.331330+00:00"},{"alias_kind":"arxiv_version","alias_value":"1703.05421v1","created_at":"2026-05-18T00:48:34.331330+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.05421","created_at":"2026-05-18T00:48:34.331330+00:00"},{"alias_kind":"pith_short_12","alias_value":"QWOZGRCQDS2V","created_at":"2026-05-18T12:31:39.905425+00:00"},{"alias_kind":"pith_short_16","alias_value":"QWOZGRCQDS2VITV5","created_at":"2026-05-18T12:31:39.905425+00:00"},{"alias_kind":"pith_short_8","alias_value":"QWOZGRCQ","created_at":"2026-05-18T12:31:39.905425+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/QWOZGRCQDS2VITV5WPKOPRFPE4","json":"https://pith.science/pith/QWOZGRCQDS2VITV5WPKOPRFPE4.json","graph_json":"https://pith.science/api/pith-number/QWOZGRCQDS2VITV5WPKOPRFPE4/graph.json","events_json":"https://pith.science/api/pith-number/QWOZGRCQDS2VITV5WPKOPRFPE4/events.json","paper":"https://pith.science/paper/QWOZGRCQ"},"agent_actions":{"view_html":"https://pith.science/pith/QWOZGRCQDS2VITV5WPKOPRFPE4","download_json":"https://pith.science/pith/QWOZGRCQDS2VITV5WPKOPRFPE4.json","view_paper":"https://pith.science/paper/QWOZGRCQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1703.05421&json=true","fetch_graph":"https://pith.science/api/pith-number/QWOZGRCQDS2VITV5WPKOPRFPE4/graph.json","fetch_events":"https://pith.science/api/pith-number/QWOZGRCQDS2VITV5WPKOPRFPE4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QWOZGRCQDS2VITV5WPKOPRFPE4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QWOZGRCQDS2VITV5WPKOPRFPE4/action/storage_attestation","attest_author":"https://pith.science/pith/QWOZGRCQDS2VITV5WPKOPRFPE4/action/author_attestation","sign_citation":"https://pith.science/pith/QWOZGRCQDS2VITV5WPKOPRFPE4/action/citation_signature","submit_replication":"https://pith.science/pith/QWOZGRCQDS2VITV5WPKOPRFPE4/action/replication_record"}},"created_at":"2026-05-18T00:48:34.331330+00:00","updated_at":"2026-05-18T00:48:34.331330+00:00"}