# Is torus symmetry forced? (O5)

- kind: question
- status: open
- year: 2026
- authors: D. Fong and Claude Fable 5 (claims ledger, July 2026)
- in one line: O5 — must every Keller counterexample carry a torus symmetry?

**Statement.** Is every non-automorphic Keller map of $\mathbb C^n$, after a polynomial change of coordinates, equivariant for a nontrivial $\mathbb G_m$-action?

Alpöge's map, Gallagher's family, the factory maps and Harish's map are all graded by a torus with weights of mixed sign. Is that forced?

The evidence is local. At Alpöge's map and at Gallagher's $F_4$, the Fong–Fable ledger reports the first-order Keller deformations of degree at most 7. Every one is either a composition move or an equivariant deformation. At $F$ a single residual direction is obstructed at second order, and none remains at $F_4$. The ledger calls this local, capped and two points deep, not a classification.[1]

Why it matters: in the plane, [Shaska's theorem](/wiki/Statement:graded-equivariant-keller.md) makes every torus-symmetric Keller map an automorphism. If O5 holds in dimension two, there is no plane counterexample, so [a positive answer to O5 would prove JC₂, through Shaska's plane theorem (itself unrefereed)](/wiki/Statement:jc-plane.md). A negative answer in dimension three would be a counterexample by a genuinely new mechanism.

## Relations

- implies [jc-plane](/wiki/Statement:jc-plane.md) — If O5 holds in dimension two, there is no plane counterexample, so a positive answer to O5 would prove JC₂, through Shaska's plane theorem (itself unrefereed).

## References

1. D. Fong and Claude Fable 5, theorems register (theorem 8) and [claims ledger](https://jacobianconjectures.com/CLAIMS.md), jacobianconjectures.com.

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