{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:K5NAKZJL53434NAFUOXCWBBAQK","short_pith_number":"pith:K5NAKZJL","schema_version":"1.0","canonical_sha256":"575a05652beef9be3405a3ae2b04208287bdc7a9fb753527701ba42428d2e239","source":{"kind":"arxiv","id":"2607.01569","version":1},"attestation_state":"computed","paper":{"title":"Complete Integrability for Piecewise-Smooth Distributions","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"math.DG","authors_text":"Jack McKee","submitted_at":"2026-07-02T00:59:03Z","abstract_excerpt":"Two generalizations of the Frobenius integrability theorem are proved concerning distributions which are piecewise-C1 but may fail to be continuous. The criteria presented are sufficient, but not necessary, for complete integrability of such distributions with bi-Lipschitz coordinates."},"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":"2607.01569","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.DG","submitted_at":"2026-07-02T00:59:03Z","cross_cats_sorted":["math.OC"],"title_canon_sha256":"44c140ce58ef2232e39590cf84676c43e8933ed2b01845826df2be50beea4029","abstract_canon_sha256":"7427d8ca9311ee2705829ce8ce296a24d4f15006bf4b1afd2e4a8e9dddfe0c14"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-03T01:17:02.183768Z","signature_b64":"zU6gvSU57mSHS69OOQBdUJNuVM8cphZSfDe8OIFhtUpQ6IBWubah1+0/DdS6yxr57xX98HpSe3ukG/I3lAqoDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"575a05652beef9be3405a3ae2b04208287bdc7a9fb753527701ba42428d2e239","last_reissued_at":"2026-07-03T01:17:02.183255Z","signature_status":"signed_v1","first_computed_at":"2026-07-03T01:17:02.183255Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Complete Integrability for Piecewise-Smooth Distributions","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"math.DG","authors_text":"Jack McKee","submitted_at":"2026-07-02T00:59:03Z","abstract_excerpt":"Two generalizations of the Frobenius integrability theorem are proved concerning distributions which are piecewise-C1 but may fail to be continuous. The criteria presented are sufficient, but not necessary, for complete integrability of such distributions with bi-Lipschitz coordinates."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.01569","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.01569/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2607.01569","created_at":"2026-07-03T01:17:02.183309+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.01569v1","created_at":"2026-07-03T01:17:02.183309+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.01569","created_at":"2026-07-03T01:17:02.183309+00:00"},{"alias_kind":"pith_short_12","alias_value":"K5NAKZJL5343","created_at":"2026-07-03T01:17:02.183309+00:00"},{"alias_kind":"pith_short_16","alias_value":"K5NAKZJL53434NAF","created_at":"2026-07-03T01:17:02.183309+00:00"},{"alias_kind":"pith_short_8","alias_value":"K5NAKZJL","created_at":"2026-07-03T01:17:02.183309+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/K5NAKZJL53434NAFUOXCWBBAQK","json":"https://pith.science/pith/K5NAKZJL53434NAFUOXCWBBAQK.json","graph_json":"https://pith.science/api/pith-number/K5NAKZJL53434NAFUOXCWBBAQK/graph.json","events_json":"https://pith.science/api/pith-number/K5NAKZJL53434NAFUOXCWBBAQK/events.json","paper":"https://pith.science/paper/K5NAKZJL"},"agent_actions":{"view_html":"https://pith.science/pith/K5NAKZJL53434NAFUOXCWBBAQK","download_json":"https://pith.science/pith/K5NAKZJL53434NAFUOXCWBBAQK.json","view_paper":"https://pith.science/paper/K5NAKZJL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.01569&json=true","fetch_graph":"https://pith.science/api/pith-number/K5NAKZJL53434NAFUOXCWBBAQK/graph.json","fetch_events":"https://pith.science/api/pith-number/K5NAKZJL53434NAFUOXCWBBAQK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/K5NAKZJL53434NAFUOXCWBBAQK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/K5NAKZJL53434NAFUOXCWBBAQK/action/storage_attestation","attest_author":"https://pith.science/pith/K5NAKZJL53434NAFUOXCWBBAQK/action/author_attestation","sign_citation":"https://pith.science/pith/K5NAKZJL53434NAFUOXCWBBAQK/action/citation_signature","submit_replication":"https://pith.science/pith/K5NAKZJL53434NAFUOXCWBBAQK/action/replication_record"}},"created_at":"2026-07-03T01:17:02.183309+00:00","updated_at":"2026-07-03T01:17:02.183309+00:00"}