# Graded Keller maps

- kind: theorem
- status: proved (evidence tier V4)
- year: 2026
- authors: T. Shaska (arXiv:2607.20210, 22 and 25 July 2026); the plane case independently by F. Santibañez-Leal (Zenodo, September 2026)
- in one line: One-sign torus weights force an automorphism; in the plane, any symmetry does

**Statement.** Let $\mathbb G_m$ act diagonally on source and target with integer weights $\mathbf w,\mathbf q$, and let $G$ be an equivariant Keller map. (i) If all $w_i$ are nonzero and of one sign, $G$ is an automorphism. (ii) If $n=2$ and $G$ is equivariant for any nontrivial algebraic $\mathbb G_m$-action, diagonal or not, $G$ is an automorphism.

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](/wiki/Statement:jc-n.md). 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](/wiki/Statement:proper-etale-bijective.md).
- **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](/wiki/Statement:jc-plane.md).
- **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](/wiki/Statement:equivariance-forced-question.md).

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](/wiki/Statement:master-equation.md) of the Fong–Fable ledger, derived independently.

## Relations

- special case of [jc-n](/wiki/Statement:jc-n.md) — So the Jacobian conjecture holds for every Keller map graded with one-sign weights, in every dimension.
- uses [proper-etale-bijective](/wiki/Statement:proper-etale-bijective.md) — Weighted scaling then makes it proper, and a proper Keller map is an automorphism.
- special case of [jc-plane](/wiki/Statement:jc-plane.md) — So the plane conjecture holds for every Jacobian pair with a nontrivial torus symmetry.

## References

1. T. Shaska, [Graded Keller maps and the Jacobian Conjecture](https://arxiv.org/abs/2607.20210), arXiv:2607.20210 (v1 22 July, v2 25 July 2026). Unrefereed.
2. D. Fong and Claude Fable 5, [claims ledger](https://jacobianconjectures.com/CLAIMS.md), “Plane obstruction” (20 July 2026).
3. F. Santibañez-Leal, [The Jacobian Conjecture after the Three-Dimensional Counterexample](https://zenodo.org/records/22834207), Zenodo, v0.13, 18 September 2026.

## Relations to this page

- related to [harish-k4](/wiki/Statement:harish-k4.md) — The target weights are a permutation of the source weights, as Shaska's theorem requires of every graded Keller map, and they have mixed signs, as it requires of every graded counterexample.
- related to [master-equation](/wiki/Statement:master-equation.md) — Shaska derives the same order-two vanishing on the contracted locus independently, as a bracket identity on the quotient (Thm 8.3). The equation is the tool behind the plane wall for torus-symmetric maps.

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