{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:VG7WSPIN7STDIVGT2TGZXJGSYT","short_pith_number":"pith:VG7WSPIN","schema_version":"1.0","canonical_sha256":"a9bf693d0dfca63454d3d4cd9ba4d2c4cc22b3c1c57a2994918ddc39722d0aad","source":{"kind":"arxiv","id":"2607.16171","version":1},"attestation_state":"computed","paper":{"title":"A Globally Asymptotically Stable Planar Homogeneous Polynomial Vector Field With No Polynomial Lyapunov Function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.SY","eess.SY","math.OC"],"primary_cat":"math.DS","authors_text":"Jun Liu, Maxwell Fitzsimmons","submitted_at":"2026-07-17T17:50:02Z","abstract_excerpt":"We disprove the conjecture that every globally asymptotically stable homogeneous polynomial vector field admits a homogeneous polynomial Lyapunov function. The counterexample is a planar homogeneous cubic polynomial vector field with integer coefficients. It admits no positive definite homogeneous polynomial with nonpositive Lie derivative and, more strongly, no real-analytic Lyapunov function even locally. Nevertheless, it has an explicit degree-two homogeneous Lyapunov function that is radially unbounded, continuously differentiable everywhere, and smooth away from the origin. We also provid"},"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":"2607.16171","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-07-17T17:50:02Z","cross_cats_sorted":["cs.SY","eess.SY","math.OC"],"title_canon_sha256":"1e42a6f0fce3d22b1602b310d19e304d54ded59c44f31d9257715aee4aed7937","abstract_canon_sha256":"c5650ac939c430679b4f78bbaf3f8d3b743b78c755fb05e1b3a227787b199842"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-20T02:19:37.472714Z","signature_b64":"ntfvIvtMNjmOjQ5mcKM9lMxq/GmIR7Bfr+9ANJTzK7ABqqlN2R/6j24lXjd/YbEXsLWreuQuffdlO2G2vC1mCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a9bf693d0dfca63454d3d4cd9ba4d2c4cc22b3c1c57a2994918ddc39722d0aad","last_reissued_at":"2026-07-20T02:19:37.471756Z","signature_status":"signed_v1","first_computed_at":"2026-07-20T02:19:37.471756Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Globally Asymptotically Stable Planar Homogeneous Polynomial Vector Field With No Polynomial Lyapunov Function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.SY","eess.SY","math.OC"],"primary_cat":"math.DS","authors_text":"Jun Liu, Maxwell Fitzsimmons","submitted_at":"2026-07-17T17:50:02Z","abstract_excerpt":"We disprove the conjecture that every globally asymptotically stable homogeneous polynomial vector field admits a homogeneous polynomial Lyapunov function. The counterexample is a planar homogeneous cubic polynomial vector field with integer coefficients. It admits no positive definite homogeneous polynomial with nonpositive Lie derivative and, more strongly, no real-analytic Lyapunov function even locally. Nevertheless, it has an explicit degree-two homogeneous Lyapunov function that is radially unbounded, continuously differentiable everywhere, and smooth away from the origin. We also provid"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.16171","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/2607.16171/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":"2607.16171","created_at":"2026-07-20T02:19:37.472204+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.16171v1","created_at":"2026-07-20T02:19:37.472204+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.16171","created_at":"2026-07-20T02:19:37.472204+00:00"},{"alias_kind":"pith_short_12","alias_value":"VG7WSPIN7STD","created_at":"2026-07-20T02:19:37.472204+00:00"},{"alias_kind":"pith_short_16","alias_value":"VG7WSPIN7STDIVGT","created_at":"2026-07-20T02:19:37.472204+00:00"},{"alias_kind":"pith_short_8","alias_value":"VG7WSPIN","created_at":"2026-07-20T02:19:37.472204+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/VG7WSPIN7STDIVGT2TGZXJGSYT","json":"https://pith.science/pith/VG7WSPIN7STDIVGT2TGZXJGSYT.json","graph_json":"https://pith.science/api/pith-number/VG7WSPIN7STDIVGT2TGZXJGSYT/graph.json","events_json":"https://pith.science/api/pith-number/VG7WSPIN7STDIVGT2TGZXJGSYT/events.json","paper":"https://pith.science/paper/VG7WSPIN"},"agent_actions":{"view_html":"https://pith.science/pith/VG7WSPIN7STDIVGT2TGZXJGSYT","download_json":"https://pith.science/pith/VG7WSPIN7STDIVGT2TGZXJGSYT.json","view_paper":"https://pith.science/paper/VG7WSPIN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.16171&json=true","fetch_graph":"https://pith.science/api/pith-number/VG7WSPIN7STDIVGT2TGZXJGSYT/graph.json","fetch_events":"https://pith.science/api/pith-number/VG7WSPIN7STDIVGT2TGZXJGSYT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VG7WSPIN7STDIVGT2TGZXJGSYT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VG7WSPIN7STDIVGT2TGZXJGSYT/action/storage_attestation","attest_author":"https://pith.science/pith/VG7WSPIN7STDIVGT2TGZXJGSYT/action/author_attestation","sign_citation":"https://pith.science/pith/VG7WSPIN7STDIVGT2TGZXJGSYT/action/citation_signature","submit_replication":"https://pith.science/pith/VG7WSPIN7STDIVGT2TGZXJGSYT/action/replication_record"}},"created_at":"2026-07-20T02:19:37.472204+00:00","updated_at":"2026-07-20T02:19:37.472204+00:00"}