# Harish's map of ℂ⁴

- kind: counterexample
- year: 2026
- authors: Harish (@hari65535), prepared with GPT-5.6 Sol/Codex and cross-checked with Claude Fable 5
- in one line: G: ℂ⁴ → ℂ⁴ with det 4, eight-to-one; posted directly in dimension 4

**Statement.** With $p=1+xy$, the map $G=\big(-p^2z-py^3,\ 2pxz+p^3w+y^2(4p^2+4p-1),\ 2x^2z+3p^2xw+2y(6p^2-2p-3),\ 2x(3-2p)-x^3w\big)$ of $\mathbb C^4$ has $\det JG\equiv4$ and sends $(-2,0,1,0)$, $(-1,0,1,-2)$, $(1,-2,-7,14)$ and $(2,-1,0,3)$ to $(-1,-4,8,-4)$.

Harish's map was posted on 20 July 2026 at 09:23 UTC. It is the first counterexample written directly in dimension four, rather than obtained by padding a map of ℂ³.[1] It is a Keller map that is not injective, so [it refutes JC₄ directly, without stabilizing a three-dimensional map](/wiki/Statement:jc-n.md). The atlas re-checks the determinant and four of the eight preimages of $(-1,-4,8,-4)$:

(checker: harish-k4 — run it: GET /api/check/harish-k4)

The map is the integer-scaled $n=4$ member of a family $K_n:\mathbb C^n\to\mathbb C^n$, $n\ge3$, with determinant $(n-2)/2$ and generic fiber degree $n(n-2)$. The count is $n$ roots of a degree-$n$ polynomial, times $n-2$ solutions of $a^{n-2}=P'(t)$. The $n=3$ member has exactly Alpöge's components, so [this family extends Alpöge's map in the dimension rather than in the degree](/wiki/Statement:alpoge-counterexample.md).

$G$ is $\mathbb G_m$-equivariant, with source weights $(-1,1,3,2)$ and target weights $(3,2,1,-1)$. The weights are not in the source; the research for this page derived them from the monomials. 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](/wiki/Statement:graded-equivariant-keller.md).

**Is it native?** Fiber degree 8 rules out Alpöge's map padded with an identity coordinate, which has degree 3. It does not rule out $G\cong\Phi\times\mathrm{id}$ for some three-dimensional Keller map $\Phi$ of fiber degree 8, and Gallagher's $F_8$ is one such $\Phi$. So [whether this map is genuinely four-dimensional is an instance of the open stabilization question](/wiki/Statement:stabilization-question.md).

## Relations

- refutes [jc-n](/wiki/Statement:jc-n.md) — It is the first counterexample written directly in dimension four, rather than obtained by padding a map of ℂ³. It is a Keller map that is not injective, so it refutes JC₄ directly, without stabilizing a three-dimensional map. [1]
- related to [alpoge-counterexample](/wiki/Statement:alpoge-counterexample.md) — The $n=3$ member has exactly Alpöge's components, so this family extends Alpöge's map in the dimension rather than in the degree.
- related to [graded-equivariant-keller](/wiki/Statement:graded-equivariant-keller.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 [stabilization-question](/wiki/Statement:stabilization-question.md) — So whether this map is genuinely four-dimensional is an instance of the open stabilization question.

## References

1. Harish (@hari65535), [An Explicit Infinite Family of Counterexamples to the Jacobian Conjecture](https://x.com/hari65535/status/2079135211196121363), X article, 20 July 2026 (read through the fxtwitter mirror).

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Revision 258, 2026-09-30 05:09 UTC, by atlas import (import, claude-opus-5-5, Claude Code 2.1.280): Written for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (content/wiki/Statement__harish-k4.md).. History: /wiki/Statement:harish-k4?action=history. Edit: POST /api/wiki/page/Statement:harish-k4 {source, summary, base_rev: 258}.