{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:6YVS3CXNX66MUFPEVF3G46SVR3","short_pith_number":"pith:6YVS3CXN","schema_version":"1.0","canonical_sha256":"f62b2d8aedbfbcca15e4a9766e7a558ee3f02b7a4e90a648190a9631d005309a","source":{"kind":"arxiv","id":"2508.20241","version":1},"attestation_state":"computed","paper":{"title":"$b^k$-algebroids and the variety of foliation jets","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.DG","authors_text":"Aldo Witte, \\'Alvaro del Pino, Francis Bischoff","submitted_at":"2025-08-27T19:51:50Z","abstract_excerpt":"We introduce and classify singular foliations of $b^{k+1}$-type, which formalize the properties of vector fields that are tangent to a submanifold $W \\subset M$ to order $k$. When $W$ is a hypersurface, these structures are Lie algebroids generalizing the $b^{k+1}$-tangent bundles introduced by Scott.\n  We prove that singular foliations of $b^{k+1}$-type are encoded by $k$-th order foliations: jets of distributions that are involutive up to order $k$, equivalently described as foliations on the $k$-th order neighborhood of $W$. Using this encoding, we construct topological groupoids of $k$-th "},"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":"2508.20241","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2025-08-27T19:51:50Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"e84ae2b83dd8968b9e61581b87e701db22d0bbc0579614e83ba852de0160a78f","abstract_canon_sha256":"4a50d408c81ac2326b9dfb4c1b612589c9273e97d952f9bb1ebd4304f41b1ba6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T12:00:32.499971Z","signature_b64":"YWbyXIGa+wgKUsBREoKvcbmkcEc8UT14scaJYBGtnTCHPMdXEPq9onxW2vf+IRckREZMqPu9nr0pKZ8TAxpNBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f62b2d8aedbfbcca15e4a9766e7a558ee3f02b7a4e90a648190a9631d005309a","last_reissued_at":"2026-07-05T12:00:32.499447Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T12:00:32.499447Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$b^k$-algebroids and the variety of foliation jets","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.DG","authors_text":"Aldo Witte, \\'Alvaro del Pino, Francis Bischoff","submitted_at":"2025-08-27T19:51:50Z","abstract_excerpt":"We introduce and classify singular foliations of $b^{k+1}$-type, which formalize the properties of vector fields that are tangent to a submanifold $W \\subset M$ to order $k$. When $W$ is a hypersurface, these structures are Lie algebroids generalizing the $b^{k+1}$-tangent bundles introduced by Scott.\n  We prove that singular foliations of $b^{k+1}$-type are encoded by $k$-th order foliations: jets of distributions that are involutive up to order $k$, equivalently described as foliations on the $k$-th order neighborhood of $W$. Using this encoding, we construct topological groupoids of $k$-th "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.20241","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/2508.20241/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":"2508.20241","created_at":"2026-07-05T12:00:32.499510+00:00"},{"alias_kind":"arxiv_version","alias_value":"2508.20241v1","created_at":"2026-07-05T12:00:32.499510+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2508.20241","created_at":"2026-07-05T12:00:32.499510+00:00"},{"alias_kind":"pith_short_12","alias_value":"6YVS3CXNX66M","created_at":"2026-07-05T12:00:32.499510+00:00"},{"alias_kind":"pith_short_16","alias_value":"6YVS3CXNX66MUFPE","created_at":"2026-07-05T12:00:32.499510+00:00"},{"alias_kind":"pith_short_8","alias_value":"6YVS3CXN","created_at":"2026-07-05T12:00:32.499510+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/6YVS3CXNX66MUFPEVF3G46SVR3","json":"https://pith.science/pith/6YVS3CXNX66MUFPEVF3G46SVR3.json","graph_json":"https://pith.science/api/pith-number/6YVS3CXNX66MUFPEVF3G46SVR3/graph.json","events_json":"https://pith.science/api/pith-number/6YVS3CXNX66MUFPEVF3G46SVR3/events.json","paper":"https://pith.science/paper/6YVS3CXN"},"agent_actions":{"view_html":"https://pith.science/pith/6YVS3CXNX66MUFPEVF3G46SVR3","download_json":"https://pith.science/pith/6YVS3CXNX66MUFPEVF3G46SVR3.json","view_paper":"https://pith.science/paper/6YVS3CXN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2508.20241&json=true","fetch_graph":"https://pith.science/api/pith-number/6YVS3CXNX66MUFPEVF3G46SVR3/graph.json","fetch_events":"https://pith.science/api/pith-number/6YVS3CXNX66MUFPEVF3G46SVR3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6YVS3CXNX66MUFPEVF3G46SVR3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6YVS3CXNX66MUFPEVF3G46SVR3/action/storage_attestation","attest_author":"https://pith.science/pith/6YVS3CXNX66MUFPEVF3G46SVR3/action/author_attestation","sign_citation":"https://pith.science/pith/6YVS3CXNX66MUFPEVF3G46SVR3/action/citation_signature","submit_replication":"https://pith.science/pith/6YVS3CXNX66MUFPEVF3G46SVR3/action/replication_record"}},"created_at":"2026-07-05T12:00:32.499510+00:00","updated_at":"2026-07-05T12:00:32.499510+00:00"}