{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2026-07-24-hessian-conjecture-counterexample/","md":"https://postcutoff.com/e/2026-07-24-hessian-conjecture-counterexample/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-07-24-hessian-conjecture-counterexample","date":"2026-07-24","date_precision":"day","short_title":"Hessian conjecture refuted in five variables, derived from Claude-found Jacobian counterexample","deck":null,"takeaway":"Five days after Levent Alpöge's Claude Fable 5-assisted counterexample to the Jacobian conjecture, Guowu Meng and Liang Yang used \"Schur descent\" on it to build a five-variable counterexample to the related Hessian conjecture.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2607.22198: A five-variable counterexample to the Hessian conjecture","url":"https://arxiv.org/abs/2607.22198","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"Terence Tao: A digestion of the Jacobian conjecture counterexample","url":"https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/","type":"discussion","group":"reactions","domain":"terrytao.wordpress.com"}],"official":1,"filed":"2026-09-29","updated":"2026-10-19","orgs":["Independent researchers"],"title":"Hessian conjecture refuted in five variables, derived from Claude-found Jacobian counterexample","summary":"Five days after Levent Alpöge's Claude Fable 5-assisted counterexample to the Jacobian conjecture, Guowu Meng and Liang Yang used \"Schur descent\" on it to build a five-variable counterexample to the related Hessian conjecture. The Hessian conjecture now holds for n≤3, fails for n≥5, and is open only for n=4.","key_facts":["arXiv 2607.22198, submitted 2026-07-24 (revised 07-27)","Explicit polynomial in 5 variables, degree 14, constant Hessian determinant 128, with non-injective gradient","Derived from Alpöge's Jacobian counterexample; the paper itself does not report AI use"],"key_numbers":[],"tags":["mathematics","algebraic-geometry","follow-on-result"],"science":{"field":"mathematics","subfield":"algebraic geometry / polynomial maps","problem":"Hessian conjecture: a polynomial whose Hessian determinant is a nonzero constant has an injective gradient map","result":"Counterexample for n=5 (hence all n≥5); status now: true for n≤3, false for n≥5, open for n=4","open_since":"","ai_system":["Claude Fable 5 (indirectly, via the Jacobian counterexample)"],"human_role":"Human-led; built by hand on an AI-assisted result","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed arXiv preprint (2607.22198); explicit counterexample, checkable by computation; the paper reports no AI use"},"status":"pending","shock":"An AI-found counterexample spread to a neighbouring conjecture within days."},"body_md":"## What happened\n\nGuowu Meng and Liang Yang turned Alpöge's three-variable Jacobian counterexample into a five-variable counterexample to the Hessian conjecture.\n\n## Why it matters\n\nIt shows how AI-found results feed quickly into human follow-up work. It also leaves one clean open case, n=4.","disputed":[{"label":"Verification","text":"Unrefereed arXiv preprint (2607.22198); explicit counterexample, checkable by computation; the paper reports no AI use","sources":[]}],"related":[{"id":"2026-07-20-jacobian-conjecture-counterexample","url":"https://postcutoff.com/e/2026-07-20-jacobian-conjecture-counterexample/","date":"2026-07-20","date_precision":"day","short_title":"Claude Fable 5 finds a counterexample to the Jacobian conjecture in dimension 3","deck":null,"takeaway":"The Jacobian conjecture is one of the most famous open problems in algebra.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":9,"official":6,"filed":"2026-09-29","updated":"2026-11-10","orgs":["Anthropic"]}],"people":[{"id":"terence-tao","name":"Terence Tao","url":"https://postcutoff.com/person/terence-tao/"},{"id":"levent-alpoge","name":"Levent Alpöge","url":"https://postcutoff.com/person/levent-alpoge/"}],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-19","type":"edit","text":"Fixed ai_system (an unquoted comma split it into two items); verification made precise"},{"date":"2026-09-29","type":"filed","text":"Created. During a snowball check while verifying the Jacobian entry"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-09-29","run":null,"sources_read":null,"updated":"2026-10-19","human_review":null,"version":{"date":"2026-10-19"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":85,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":24,"in_training_data":false},{"model_id":"gemini-4-hydrogen","name":"Gemini 4 Hydrogen","cutoff":"2026-03","days_after":115,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":54,"in_training_data":false}],"short_url":null}