The master equation

Alpöge's map is equivariant for the torus, so it is determined by what it does on the invariants $t=xy$ and $w=x^2z$. On 20 July 2026 the Fong–Fable session computed that descent, as follows.^[1]

• The products $F_2F_3$ and $F_1F_3^2$ are invariant.
• With $U=1+t$, $S=wU^2+3t^2U+t^2$, $\tilde g=t+3S$, $\tilde h=US$ and $m=2-3t-w$, they equal $m\tilde g$ and $m^2\tilde h$.
• The constant Jacobian upstairs becomes a single identity in the Poisson bracket $\{f,g\}=f_tg_w-f_wg_t$, with constant $c=2$.

The atlas re-checks all three identities exactly on every view:

The ledger reports the same identity for Gallagher's $F_4$, with $c=-1$, and for the factory's branch N2. The atlas has not re-checked those two. The general equivalence is a chain-rule computation through the descent. It is unrefereed, hence V4. Shaska derives the same order-two vanishing on the contracted locus independently, as a bracket identity on the quotient (Thm 8.3).^[2] The equation is the tool behind the plane wall for torus-symmetric maps. In two variables the torus has a one-dimensional invariant ring, the equation degenerates to $g'(t)=c$, and only affine maps survive.

It is also the setting of the ledger's open question O4: quantize the master equation to get an explicit non-surjective endomorphism of the Weyl algebra $A_3$. An explicit one already exists by the classical pullback (see DCₙ).

1. ↑ D. Fong and Claude Fable 5, claims ledger, entry C5 and theorem 5 of the theorems register.
2. ↑ T. Shaska, Graded Keller maps and the Jacobian Conjecture, Theorem 8.3.