# Zyskind's factory

- kind: construction
- year: 2026
- authors: Guy Zyskind (@GuyZys), crediting GPT-5.6 Sol
- in one line: A claimed parametric generator of Keller maps of ℂ³ (A = 1 + xyᵐ)

**Statement.** For integers $m,r\ge1$, $C\ne0$ and a polynomial $h$, put $A=1+xy^m$, $B=A^{r+1}z+y^{m+1}h(A)$, $P=AB$, $Q=y+xB$ and $R=C\int_0^{x/A}\big(1-t(Q-Pt)^m\big)^r\,dt$. Claimed: when $h$ makes $R$ a polynomial, $(P,Q,R)$ has $\det J\equiv-C$ and generic fiber degree $r(m+1)+1$.

Posted on 20 July 2026 at 22:40 UTC under the line “the jacobian conjecture is not merely false, it is factory false”. It lists 16 maps said to be exactly checked: the seed (Alpöge's map), nine base maps from three branches, three compositions in ℂ³ and three products into ℂ⁶ and ℂ⁹. Some have coefficients in a number field. The example N2 has $(m,r,C)=(2,1,12)$ and $h=u+u^2A$ with $u^2-4u+6=0$, so $u\in\mathbb Q(\sqrt{-2})$, and fiber degree 4.[1]

If the claim holds, [every factory output is a Keller map of ℂ³ with at least three preimages over a general point, and so refutes JC₃](/wiki/Statement:jc-3.md). The general claim has no written proof. The Fong–Fable ledger reports re-verifying branch N2 (its denominators cancel modulo $u^2-4u+6$, and $\det=-12=-C$) and leaves the other 15 maps unverified.[2] The research for this page checked the seed by hand. At $(m,r)=(1,1)$, $h(A)=3+9A$, the factory map is Alpöge's after $z\mapsto3z$ and rescaling outputs, so [the factory's first member is Alpöge's map](/wiki/Statement:alpoge-counterexample.md).

From the formulas, every base map is $\mathbb G_m$-equivariant with source weights $(m,-1,-(m+1))$. That includes the number-field members, so none of them is a counterexample without torus symmetry.

## Relations

- refutes [jc-3](/wiki/Statement:jc-3.md) — If the claim holds, every factory output is a Keller map of ℂ³ with at least three preimages over a general point, and so refutes JC₃.
- related to [alpoge-counterexample](/wiki/Statement:alpoge-counterexample.md) — At $(m,r)=(1,1)$, $h(A)=3+9A$, the factory map is Alpöge's after $z\mapsto3z$ and rescaling outputs, so the factory's first member is Alpöge's map.

## References

1. G. Zyskind, [X post of 20 July 2026](https://x.com/GuyZys/status/2079335737829580957) (read through the fxtwitter mirror).
2. D. Fong and Claude Fable 5, [claims ledger](https://jacobianconjectures.com/CLAIMS.md).

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Revision 285, 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__zyskind-factory.md).. History: /wiki/Statement:zyskind-factory?action=history. Edit: POST /api/wiki/page/Statement:zyskind-factory {source, summary, base_rev: 285}.