No known construction converts the three-dimensional Jacobian counterexamples into a counterexample over the complex plane
3 events · 1 assessment · 1 decision
Structured and assessed
First pass. Canonical form amended to "over the complex plane": the unqualified wording is false in characteristic two, where Mondello's July 2026 planar counterexample to the separable Jacobian conjecture was derived as a fibre of Huq-Kuruvilla's three-variable map (arXiv:2608.02634); the parent claim and every source have the characteristic-zero plane in view, so this fixes scope without changing identity or direction. Decomposition: two named arguments. "Proved obstruction to graded descent" attaches the existing claims on Shaska's equivariant-Keller theorem (40e7acae) and Vitushkin's obstruction (7230e581), both found via similarity search after the Matcher timed out on a proposed "tangent-sweep requires dimension three" subclaim; those existing nodes carry the same content, so nothing new was minted there. "The plane is uniformly reported untouched" attaches the existing upward-propagation claim (65193a2e) and one new subclaim, "The two-dimensional Jacobian conjecture remains open" (446ad595), which the Matcher confirmed novel at 0.92 and which is heavily consulted; seeded at 0.97. The failure in every dimension at least three (a87e655a) attached as assumes. Assessment: verified, confidence 0.85, credence 0.95, marginal yield 0.15. Five affirming instances recorded from sources read this pass (Gao, Shaska, Meng–Yang, MathWorld, an anonymous machine-assisted comment on Tao's thread); none denies. Provenance readings recorded for all five, one edge (MathWorld's planar clause cites Zhang's note, which discusses only dimensions three and above), and a source map marked material because most instances are status reports while the substantive support is Shaska's theorem and the slice checks. Importance set to 0.3, contestation 0.25. Tool notes: Gao's PDF failed on encoding (HTML read instead); the Lovely Little Lemmas post returned 403. The parent (5f9a607f) is notified since this claim moved from unassessed to verified and its wording gained a scope qualifier.
Assessed Verified
verdict confidence 0.85 · credence 0.95
The Jacobian conjecture was refuted in three variables in July 2026, and a counterexample in any dimension lifts to every higher dimension by adjoining identity coordinates. Nothing runs the other way. Restricting or projecting a three-variable Keller map does not in general preserve the Keller condition, and as of September 2026 no one has exhibited a reduction that carries any of the known counterexamples down to the complex plane. Every account of the refutation, from Terence Tao's expository post and Shuhong Gao's survey of the tangent-sweep construction to the Meng and Yang status note and reference entries updated since, records the two-variable complex case as open and untouched. The absence is not merely a matter of nobody having tried. All known three-dimensional counterexamples share a mechanism: a many-to-one enveloping map whose Jacobian is a coordinate, followed by a monomial twist that cancels that coordinate and pushes the ramification off to infinity. The resulting maps are homogeneous for a mixed-sign scaling of their coordinates, and every scaling-equivariant polynomial Keller map of the plane is an automorphism: in two variables the Keller condition descends to an identity between one-variable polynomials in which the Jacobian is a derivative, and a degree count closes the case. Any descent preserving the symmetry of the known examples is therefore impossible, and the two-variable version of the construction, tried by Vitushkin in 1999, yields only a rational map. Machine-checked computations reported on Tao's blog found no coordinate or linear slice of the counterexample's underlying variety that is an affine plane, with slices mixing coordinates nonlinearly left untested. Two contrasts sharpen the point. In characteristic two, a planar counterexample to the separable Jacobian conjecture was obtained in July 2026 as a fibre of a three-variable map, so a downward construction does exist there; it depends on cancellations special to that characteristic, and the natural fibres of the complex counterexample are automorphisms. And the doubled three-dimensional counterexample has been carried into the Hessian family by a one-variable Legendre descent, refuting the Hessian conjecture in five variables and above, but the known implication between the families runs from the four-variable Hessian conjecture to the planar Jacobian conjecture, not the reverse, so that route cannot reach the plane either. The claim describes the current state of methods, and a single new idea would falsify it; nothing of the kind has been announced.
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