{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2022:JFAIWLWI4T6WM5F4QKYULUSN6W","short_pith_number":"pith:JFAIWLWI","schema_version":"1.0","canonical_sha256":"49408b2ec8e4fd6674bc82b145d24df5a0da534727907db78afd2398ff13c326","source":{"kind":"arxiv","id":"2208.04913","version":1},"attestation_state":"computed","paper":{"title":"Polar Coordinates in Carnot groups II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","math.MG"],"primary_cat":"math.AP","authors_text":"Jeremy T. Tyson","submitted_at":"2022-08-09T17:15:54Z","abstract_excerpt":"A Carnot group is polarizable if it carries a homogeneous norm whose powers are fundamental solutions for the $p$-sub-Laplacian operators for all $1<p \\le \\infty$. Such groups also support a system of horizontal polar coordinates. We prove that the converse statement is true: if a Carnot group supports a horizontal polar coordinate system with suitable properties, then it is polarizable."},"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":"2208.04913","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2022-08-09T17:15:54Z","cross_cats_sorted":["math.DG","math.MG"],"title_canon_sha256":"4cb23885debfbb7ab64a8da0e9ddff82d6a54f53b2b9de2a7b5f1c64613c335d","abstract_canon_sha256":"210b1a48e9d48e369a9d7372bbebc2a8da7edb4916216d49f0cfdc1ce9b2ea5d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:56:30.940767Z","signature_b64":"/8nUEhaHYs2ODKmgIp0N4fHWsWh+hI4/scYQhBBRUOG+p59mkDeZTwmMhzso6zTC61MsHkJ3Ma2HNmpui/lUAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"49408b2ec8e4fd6674bc82b145d24df5a0da534727907db78afd2398ff13c326","last_reissued_at":"2026-07-05T10:56:30.940249Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:56:30.940249Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Polar Coordinates in Carnot groups II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","math.MG"],"primary_cat":"math.AP","authors_text":"Jeremy T. Tyson","submitted_at":"2022-08-09T17:15:54Z","abstract_excerpt":"A Carnot group is polarizable if it carries a homogeneous norm whose powers are fundamental solutions for the $p$-sub-Laplacian operators for all $1<p \\le \\infty$. Such groups also support a system of horizontal polar coordinates. We prove that the converse statement is true: if a Carnot group supports a horizontal polar coordinate system with suitable properties, then it is polarizable."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2208.04913","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/2208.04913/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":"2208.04913","created_at":"2026-07-05T10:56:30.940303+00:00"},{"alias_kind":"arxiv_version","alias_value":"2208.04913v1","created_at":"2026-07-05T10:56:30.940303+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2208.04913","created_at":"2026-07-05T10:56:30.940303+00:00"},{"alias_kind":"pith_short_12","alias_value":"JFAIWLWI4T6W","created_at":"2026-07-05T10:56:30.940303+00:00"},{"alias_kind":"pith_short_16","alias_value":"JFAIWLWI4T6WM5F4","created_at":"2026-07-05T10:56:30.940303+00:00"},{"alias_kind":"pith_short_8","alias_value":"JFAIWLWI","created_at":"2026-07-05T10:56:30.940303+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/JFAIWLWI4T6WM5F4QKYULUSN6W","json":"https://pith.science/pith/JFAIWLWI4T6WM5F4QKYULUSN6W.json","graph_json":"https://pith.science/api/pith-number/JFAIWLWI4T6WM5F4QKYULUSN6W/graph.json","events_json":"https://pith.science/api/pith-number/JFAIWLWI4T6WM5F4QKYULUSN6W/events.json","paper":"https://pith.science/paper/JFAIWLWI"},"agent_actions":{"view_html":"https://pith.science/pith/JFAIWLWI4T6WM5F4QKYULUSN6W","download_json":"https://pith.science/pith/JFAIWLWI4T6WM5F4QKYULUSN6W.json","view_paper":"https://pith.science/paper/JFAIWLWI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2208.04913&json=true","fetch_graph":"https://pith.science/api/pith-number/JFAIWLWI4T6WM5F4QKYULUSN6W/graph.json","fetch_events":"https://pith.science/api/pith-number/JFAIWLWI4T6WM5F4QKYULUSN6W/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JFAIWLWI4T6WM5F4QKYULUSN6W/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JFAIWLWI4T6WM5F4QKYULUSN6W/action/storage_attestation","attest_author":"https://pith.science/pith/JFAIWLWI4T6WM5F4QKYULUSN6W/action/author_attestation","sign_citation":"https://pith.science/pith/JFAIWLWI4T6WM5F4QKYULUSN6W/action/citation_signature","submit_replication":"https://pith.science/pith/JFAIWLWI4T6WM5F4QKYULUSN6W/action/replication_record"}},"created_at":"2026-07-05T10:56:30.940303+00:00","updated_at":"2026-07-05T10:56:30.940303+00:00"}