Graded Keller maps

Shaska found the symmetry behind the counterexample, and proved that the plane cannot have it.^[1]

• Weights must be a permutation, of mixed sign. For a graded Keller map, the target weights are a permutation of the source weights (Cor. 2.3). If all the weights are nonzero and of one sign, the map is an automorphism (Thm 3.1). So the Jacobian conjecture holds for every Keller map graded with one-sign weights, in every dimension. The proof shows that such a map sends only 0 to 0. Weighted scaling then makes it proper, and a proper Keller map is an automorphism.
• The plane admits no symmetric counterexample. In dimension two, every Keller map with any nontrivial $\mathbb G_m$-symmetry is an automorphism (Thm 3.4). The symmetry need not be diagonal. After linearizing, the map is triangular, affine, or a scaling or swap of the coordinates, depending on the signs. So the plane conjecture holds for every Jacobian pair with a nontrivial torus symmetry.
• The counterexample carries mixed signs. Alpöge's map is graded with source weights $(1,-1,-2)$ and target weights $(-2,-1,1)$. The signs are mixed and $n=3$, so the hypothesis of (i) cannot be dropped.

A graded counterexample therefore needs weights of both signs and at least three variables. Alpöge's map, Gallagher's family, the factory maps and Harish's map all have such a grading.

The Fong–Fable ledger recorded a two-line version of the plane wall for torus-equivariant twisted lifts on 20 July, and Shaska's preprint appeared on 22 July. In that version the design equation degenerates to $g'(t)=c$ and forces an affine map.^[2] Santibañez-Leal states the plane result independently for diagonal weights.^[3] Whether every counterexample must be graded is the open question O5.

The same paper proves that the Keller condition descends to the quotient by the torus as a single bracket identity (Thm 8.3). That is the master equation of the Fong–Fable ledger, derived independently.

1. ↑ T. Shaska, Graded Keller maps and the Jacobian Conjecture, arXiv:2607.20210 (v1 22 July, v2 25 July 2026). Unrefereed.
2. ↑ D. Fong and Claude Fable 5, claims ledger, “Plane obstruction” (20 July 2026).
3. ↑ F. Santibañez-Leal, The Jacobian Conjecture after the Three-Dimensional Counterexample, Zenodo, v0.13, 18 September 2026.