JC₃: the Jacobian conjecture in dimension three

JC₃ is the first dimension in which the Jacobian conjecture fails. It is the three-variable case of the general conjecture, so JC₃ is a special case of JCₙ, and its failure makes the all-dimensions conjecture false.

The failure is witnessed by Alpöge's map, announced on 19 July 2026 (US time; 20 July UTC). It is a map of total degree 7 with Jacobian determinant $-2$ that sends three points to $(-\tfrac14,0,0)$. The atlas re-checks it on every view of this page:

Padding with identity coordinates carries the failure to every higher dimension. The map $F\times\mathrm{id}_{n-3}$ has the same constant determinant and the same collision, so JCₙ fails for every $n\ge3$.

The same plane in dimension three

The walls that surround the plane case are drawn again here for ℂ³. Keller's birational theorem and Campbell's Galois theorem hold in every dimension, so rows 1 and 2 stay closed. Above them the ℂ³ plane is full of counterexamples, where the plane has none.

Other witnesses

Within a day of the announcement there were more. Gallagher's family has one counterexample of every geometric degree $n\ge3$. Zyskind's factory is a claimed parametric generator. Harish's map of ℂ⁴ is written directly in dimension four. Later came Gao's tangent sweeps in every dimension $n\ge3$.^[1] On 15 September van den Essen gave an elementary derivation that recovers Alpöge's map up to linear changes of coordinates.^[2]

Consequences

Through the classical implication from the Dixmier conjecture, the endomorphisms of the Weyl algebra $A_3$ are not all automorphisms: see DCₙ. The real version fails in dimension 3 as well, because the colliding points are rational: see the real Jacobian question. By the reductions of Bass–Connell–Wright and de Bondt–van den Essen, there are now explicit cubic-homogeneous and gradient counterexamples in higher dimensions.

Evidence

V0. The certificate is kernel-checked in Isabelle/HOL, including that the map has no polynomial inverse and that padding extends it to every $n\ge3$.^[3] The pinned formal-conjectures statement is disproved in Lean.^[4] The atlas has rebuilt neither project. Its own exact check above runs on every page view.

1. ↑ S. Gao, Counterexamples to the Jacobian conjecture in dimensions greater than two, arXiv:2608.00222 (31 July 2026).
2. ↑ A. van den Essen, An elementary way to find a counterexample to the Jacobian Conjecture, arXiv:2609.17795 (15 September 2026).
3. ↑ A. F. Ramos, D. B. Hulak, R. J. G. B. de Queiroz, Formal Verification of an Explicit Counterexample to the Jacobian Conjecture, Archive of Formal Proofs, submitted 20 July 2026.
4. ↑ google-deepmind/formal-conjectures, JacobianConjecture.lean at 6fbb54f.