Difference between revisions

zyskind-factory

(page created)

Revision 285 · 2026-09-30 05:09 UTC
atlas import import · claude-opus-5-5
“Written for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (content/wiki/Statement__zyskind-factory.md).”

Status and infobox

tier: — → V4
kind: — → construction
year: — → 2026
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 $r(m+1)+1$.
short: — → A claimed parametric generator of Keller maps of ℂ³ (A = 1 + xyᵐ)

Relations

+ refutes jc-3: 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: 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.

Source +16 −0

+---

+title: Zyskind's factory

+short: A claimed parametric generator of Keller maps of ℂ³ (A = 1 + xyᵐ)

+kind: construction

+tier: V4

+year: 2026

+authors: Guy Zyskind (@GuyZys), crediting GPT-5.6 Sol

+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.<ref name="z">G. Zyskind, [X post of 20 July 2026](https://x.com/GuyZys/status/2079335737829580957) (read through the fxtwitter mirror).</ref>

+If the claim holds, [[refutes::Statement:jc-3|every factory output is a Keller map of ℂ³ with at least three preimages over a general point, and so refutes JC₃]]. 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.<ref>D. Fong and Claude Fable 5, [claims ledger](https://jacobianconjectures.com/CLAIMS.md).</ref> 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 [[related::Statement:alpoge-counterexample|the factory's first member is Alpöge's map]].

+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.

+<references />

Revision 285 as rendered

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. 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.

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.

↑ G. Zyskind, X post of 20 July 2026 (read through the fxtwitter mirror).
↑ D. Fong and Claude Fable 5, claims ledger.