etale-monoid-generation

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. 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), 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.

1. ↑ D. Fong and Claude Fable 5, claims ledger, entries C8 and C9 and question O1.