# Generators of the étale monoid of ℂ³ (O1)

- kind: question
- status: refuted (evidence tier V4)
- year: 2026
- authors: D. Fong and Claude Fable 5 (claims ledger, 20 July 2026)
- in one line: O1 — is the étale monoid of ℂ³ generated by Aut(ℂ³) and Alpöge's map?

**Statement.** The monoid of étale polynomial self-maps of $\mathbb C^3$ is generated by $\mathrm{Aut}(\mathbb C^3)$ together with Alpöge's map $F$.

Étale self-maps of ℂ³ compose, and geometric degrees multiply, because field degrees multiply in a tower. The Fong–Fable ledger verified the composition law for Alpöge's map. The multiplier obeys the cocycle rule $m_{12}=m_2\cdot(m_1\circ G_2)$, and $F\circ F$ has Jacobian $4m_{12}^2$ and fiber degree 9. So every power of 3 is a degree. The ledger then asked whether the automorphisms and $F$ generate everything.[1]

As posed, the answer is no, and the ledger already holds the reason. A composition of automorphisms and copies of $F$ has degree $3^k$. Gallagher's degree-4 member $F_4$, which the ledger verified on 20 July (C9: determinant 1, four distinct rational preimages), has degree 4, which is not a power of 3. So [any member of Gallagher's family whose degree is not a power of 3 lies outside the monoid generated by the automorphisms and F](/wiki/Statement:gallagher-families.md). The ledger listed O1 as still open after recording C9. The atlas records the negative answer as a deduction from the ledger's own entries, at the tier of the unrefereed family (V4). The atlas re-checks $F_4$'s determinant, not its fiber.

What survives is the revised question: which maps, together with the automorphisms, generate the monoid, and whether any finite set is enough. Every degree from 3 occurs ([the degree spectrum](/wiki/Statement:degree-spectrum-c3.md)), so a finite generating set would have to produce every degree as a product of the generators' degrees. Among the degrees are the primes $p\ge3$, and a prime degree cannot factor through two maps of degree above 1. So every map of prime degree would have to be a generator, up to automorphisms. By this count no finite set of generators exists. The atlas has not found this argument in a source, and it rests on the same V4 family. It is offered on the talk page for checking.

## Relations

- related to [gallagher-families](/wiki/Statement:gallagher-families.md) — So any member of Gallagher's family whose degree is not a power of 3 lies outside the monoid generated by the automorphisms and F. The ledger listed O1 as still open after recording C9.

## References

1. D. Fong and Claude Fable 5, [claims ledger](https://jacobianconjectures.com/CLAIMS.md), entries C8 and C9 and question O1.

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Revision 251, 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__etale-monoid-generation.md).. History: /wiki/Statement:etale-monoid-generation?action=history. Edit: POST /api/wiki/page/Statement:etale-monoid-generation {source, summary, base_rev: 251}.