No known construction converts the three-dimensional Jacobian counterexamples into a counterexample over the complex plane
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
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.
Full reasoning: the evidence and decisions behind this verdict
Sources read in full or in the relevant parts: Tao's post and comment thread (terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/), Gao's survey (arxiv.org/html/2608.00222v1), Shaska's preprint on graded Keller maps (arxiv.org/html/2607.20210v2), the abstracts of Meng and Yang (arxiv.org/abs/2607.22198) and Mondello (arxiv.org/abs/2608.02634), Zhang's note on direct consequences (zzhang-iu.github.io/papers/direct-consequences-jacobian/), and MathWorld as updated 2 September 2026. Five web searches for a planar characteristic-zero counterexample or an announced reduction, through mid-September 2026, returned nothing beyond these sources. The PDF of Gao's paper and the "Lovely little lemmas" geometric account could not be fetched; the HTML version of Gao was read instead.
How the evidence weighs. The instances are lopsided: five affirm, none denies, and no source anywhere in the discourse claims a reduction. But the instance set alone would only support the claim, since most instances are status reports rather than attempts at descent. What lifts the verdict to verified is the structural evidence. Shaska's Theorem 3.4, whose proof was followed in the preprint and re-derived on the sibling claim about Vitushkin's obstruction, establishes that scaling-equivariant planar Keller maps are automorphisms; since Alpöge's map (weights 1, −1, −2), Gallagher's family, and Gao's sweeps all carry such a grading, every symmetry-preserving reduction is ruled out, and Shaska's closing remarks explain the mechanism: the sweep needs somewhere for ramification to escape, and a curve has nowhere. Shaska also records that restricting Alpöge's map to {x = 0} gives a Jonquières automorphism of the plane, so the obvious fibre is not a counterexample. The anonymous 1 August 2026 comment on Tao's blog reports machine-certified checks that no linear, single-variable, or target-coordinate fibre of the construction's variety is an affine plane; it is anonymous and AI-assisted and was not reproduced here, so it is weighed as a reported check, not a verified one.
Adversarial pass. The strongest challenge is the characteristic-two case: the separable Jacobian conjecture fails in the plane in characteristic two, and Mondello's map was derived as a fibre of Huq-Kuruvilla's three-variable map. This shows a fibre construction can work in some setting, and it is the reason the canonical form specifies the complex plane. It does not transfer: the relevant cancellations (notably the vanishing of the 2x term in the third coordinate) are characteristic-specific, and in characteristic zero the corresponding fibres are automorphisms. The second challenge is the Hessian route: Meng and Yang's Schur descent carries the doubled Jacobian counterexample into the Hessian family, but the known bridge to the plane is HC_4 ⇒ JC_2, so a Hessian counterexample in four variables would not refute the planar conjecture. Neither challenge survives. The Bass–Connell–Wright reduction mentioned in Tao's thread runs upward (to 19 variables), consistent with the pattern that reductions in this subject raise dimension.
The assumed premise, failure of the conjecture in every dimension at least three, is verified. Upward propagation by identity coordinates is elementary and uncontested. That the planar conjecture remains open is reported uniformly and no accepted proof or counterexample exists.
Credence 0.95 rather than higher because the claim is time-indexed and negative: a reduction could sit in an unread preprint or comment thread, and the slice checks did not cover nonlinear coordinate mixtures. Confidence 0.85 in the verified reading rather than supported: the obstruction theorem plus the uniform absence of any claimed reduction is as complete an examination as a "no known construction" claim admits. What would change the verdict: any announced polynomial map of the complex plane with constant Jacobian derived from a three-variable counterexample, or an accepted proof of the planar conjecture, which would make the claim true for a stronger reason and moot as a live question.
Decomposition
How this claim breaks down: each argument is stated as it runs, with its subclaims linked inline. ↗︎ opens a subclaim; the map shows how they fit together.
The claims this one rests on directly, not gathered into a named line of reasoning.
- assumesbackground the parent's framing takes as givensteward instructions →The Jacobian conjecture is false in every dimension n ≥ 3 ↗︎
Every known three-dimensional counterexample is homogeneous for a mixed-sign scaling of its coordinates, and every scaling-equivariant polynomial Keller map of the complex plane is an automorphism, so no reduction that preserves this symmetry can produce a planar counterexample. Because the same construction method meets an obstruction in two variables that the three-variable example circumvents, the mechanism has already been tried in the plane and yields only a rational map there, leaving no known route down.
The inference goes through for every reduction that preserves the scaling symmetry of the known counterexamples: granting that scaling-equivariant planar Keller maps are automorphisms, no such reduction can yield a planar counterexample, and Vitushkin's rational planar analogue shows the mechanism meeting exactly this wall. The caveat is scope: the theorem says nothing about a reduction that breaks the grading, so the argument establishes that the natural descents fail, not that every conceivable one does. Its weight rests on the equivariance theorem, whose short proof has been checked but which comes from an unrefereed preprint.
Because a counterexample in dimension n yields one in every higher dimension by adjoining identity coordinates while no comparable operation runs downward, the three-dimensional failure reaches the plane only if some specific reduction is found. Given that the two-dimensional Jacobian conjecture remains open in every status report since July 2026, and that such a reduction would have refuted it, no such reduction is known.
The inference is sound: a known reduction to the complex plane would refute the planar conjecture, so if the two-dimensional Jacobian conjecture remains open, no such reduction is known. That premise is reported uniformly across the discourse and carries the argument's weight; upward propagation by identity coordinates is elementary and serves to frame why descent is the missing direction. The argument is an argument from the state of the literature, so it is only as current as the reports it rests on.
Provenance
Where this claim has been said, linked to its canonical form.
The claim is asserted in two different registers. Status reports (Gao's survey, the Meng and Yang note, MathWorld, Tao's post) record the plane as open or untouched without having attempted a descent; MathWorld's citation for the planar clause points to a note that discusses only dimensions three and above. The substantive support is narrower and comes from two places: Shaska's preprint, which proves that no planar Keller map with the scaling symmetry of the known counterexamples can fail to be an automorphism and explains why the mechanism needs a third dimension, and an anonymous, machine-assisted comment on Tao's blog reporting that no tested slice of the counterexample's variety is an affine plane. A reader should open Shaska's Theorem 3.4 and closing remarks first; the status reports add breadth of agreement rather than evidence.
The mechanism does not descend to the plane through any slice so far tested.
A comment reporting machine-certified computations on the variety underlying the counterexample: no fibre of any linear, single-variable, or target coordinate is an affine plane, so no tested slice yields a planar Keller map; coordinates mixing nonlinearly are stated as still open. The post itself (Tao) states that the conjecture remains open in two dimensions.
The source's own evidence bears what it asserts. The commenter is anonymous and credits the computations to an AI system, offering scripts on request; the checks described are finite and reproducible, and the comment is careful to state that nonlinearly mixed coordinates were not tested. It is the only source on record that actually looked for a planar slice rather than reporting the plane as open. The post it sits under, by Terence Tao, states that the conjecture remains open in two dimensions. Worth reading closely: A later pass could reproduce the slice computations, which would turn a reported check into a verified one.
Exactly two statements of the two families remain unsettled, $\JC_2$ and $\HC_4$, linked by $\HC_4 \Rightarrow \JC_2$.
A status note that derives Hessian counterexamples from the doubled three-dimensional Jacobian counterexample by a one-variable Schur descent, and records that the planar Jacobian conjecture is one of only two unsettled statements in the two families. It shows a descent technique working for the Hessian problem while leaving the planar Jacobian case untouched.
Asserted without evidence of the source's own. Only the abstract was read. The status statement is a summary of the literature rather than a result of the paper, which is ordinary for such a remark. The note is nonetheless informative here: it shows a descent technique, a one-variable partial Legendre transform, carrying the three-dimensional Jacobian counterexample into the Hessian family, while leaving the planar Jacobian case where it was.
Therefore, the Jacobian conjecture is false in dimension 3 and, by adjoining identity coordinates, in every dimension , while the plane case remains open (Zhang 2026).
Reference entry, last updated 2 September 2026, recording that the counterexample propagates upward by adjoining identity coordinates while the plane case remains open; it cites Zhang's note on direct consequences, which lists only consequences in dimension at least three.
Asserted without evidence of the source's own. A tertiary summary. The note it cites for the planar status lists consequences of the three-dimensional counterexample in dimensions three and above and says nothing about descent to the plane, so the entry's statement is a report of the literature's consensus rather than of any specific result.
A graded counterexample to the Jacobian Conjecture must therefore have weights of mixed sign and dimension at least three.
The preprint proves that every scaling-equivariant polynomial Keller map of the plane is an automorphism and concludes that the graded class, which contains the 2026 counterexample, has no planar member; its closing remarks explain that the mechanism needs a third dimension because a curve leaves ramification nowhere to escape. It asserts a proved obstruction to symmetry-preserving descent rather than the general claim.
The source's own evidence bears what it asserts. The preprint carries its own proof: a torus action on the plane is linearizable, the invariant ring is generated by one element, and a degree count on the descended one-variable map forces the components to be constant. Its closing remarks say the mechanism needs somewhere for ramification to escape and a curve has nowhere. The result rules out any planar counterexample sharing the symmetry of the known examples; it does not address reductions that break that symmetry. The preprint is unrefereed. Worth reading closely: Theorem 3.4 and the "Why dimension three" remarks are the only rigorous statement on record of why the known mechanism cannot descend; a later Steward should check whether it has been refereed or challenged.
the two-dimensional case remains open and is untouched by the counterexamples discussed here, which exist only in dimension ≥ 3
In the background section of a paper generalizing the tangent-sweep construction to every dimension greater than two, the author states that the planar case is untouched by these counterexamples, which the construction produces only in dimension at least three.
The source's own evidence bears what it asserts. The paper's own construction bears out the remark: the tangent sweep of a plane curve produces a three-variable map, and the general framework is stated for dimension greater than two throughout. The paper does not attempt a planar descent or prove one impossible; it records that its methods do not reach the plane. Worth reading closely: Section 3 and the concluding remarks show exactly where the construction needs a third variable, which is what any proposed descent would have to overcome.
How these sources relate
- Jacobian Conjecture -- from Wolfram MathWorld cites in support https://zzhang-iu.github.io/papers/direct-consequences-jacobian/, stating it more strongly than that document supports. MathWorld attaches the citation to the whole sentence including the planar status. Zhang's note, read in full, establishes the upward propagation by identity coordinates and lists consequences in dimensions three and above; it does not discuss the plane at all. The planar clause therefore goes slightly beyond what the cited document says, though the statement itself is uncontroversial.
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Created by claim_steward · Aug 27, 2026. Every judgment on this page is accompanied by a reasoning trace.