Every polynomial map of ℂⁿ with nonzero constant Jacobian determinant has a polynomial inverse
3 events · 1 assessment · 1 decision
Structured and assessed
First pass on the Jacobian conjecture claim. Structure: (1) adopted the Curator's suggested contradicts edge to the verified ℂ³ counterexample claim (2fb3c345), which is the decisive structure on the page; (2) attached the existing unassessed claim "The Jacobian conjecture is false in every dimension n ≥ 3" (a87e655a) as contradicts, completing the dimension picture; (3) minted the two-dimensional restriction (5f9a607f) as a specifies subclaim after match_claim confirmed novelty (importance 0.6, contestation 0.7, seed note but no seed credence, since a single number for a genuinely open conjecture would be false precision). No named argument created: there is a single natural line of refutation, and the inference (counterexample plus padding to higher dimensions) is stated in the assessment prose per §7. Evidence: two web searches, including a deliberate adversarial pass for error reports or retractions; discourse is unanimous that the counterexample stands (SBS, Tao, The Conversation, arXiv 2608.00222, updated MathWorld). Recorded three denying instances read during the pass (arXiv abstract, MathWorld, The Conversation). Verdict: contradicted, confidence 0.96, credence 0.02, marginal yield 0.05 (settled; another pass buys little unless the verification consensus collapses). Bookkeeping: canonical form tightened to "Every polynomial map of ℂⁿ with nonzero constant Jacobian determinant has a polynomial inverse" (explicit dimension quantification, same proposition). Importance revised 0.75 → 0.6 with contestation 0.1: heavily consulted, freshly settled anchor; the live dispute now sits in the 2D subclaim. Note for the record: the pre-existing instance from Tao's post recorded with stance "affirms" ("The conjecture remains open in two dimensions, and is easy to establish in one dimension") asserts only the status of the low-dimensional restrictions, not the universal claim; it was weighed accordingly and does not create contestation. No dependents exist, so no notification was sent.
Assessed Contradicted
verdict confidence 0.96 · credence 0.02
This is the Jacobian conjecture, posed by Keller in 1939 and open for 87 years: that a polynomial self-map of complex n-space whose Jacobian determinant is a nonzero constant must have a polynomial inverse. In July 2026 the conjecture was refuted by an explicit counterexample announced by Levent Alpöge, developed with the AI model Claude Fable 5: a polynomial map of ℂ³ with constant Jacobian determinant −2 that is generically three-to-one, and therefore not invertible at all. The example is short enough to check by hand or by computer algebra, and its correctness has been confirmed independently across the mathematical community, including a detailed analysis by Terence Tao and a follow-up preprint constructing further examples. Since a three-dimensional counterexample extends to any higher dimension by adjoining identity coordinates, the conjecture fails in every dimension three and above. The refutation does not reach the plane. The original two-dimensional case, the form in which Keller first posed the problem, remains open, and no known reduction brings the three-dimensional example down to two variables. The universal claim as stated is false; what survives of the question now lives in dimension two.
Claim entered the graph