Write \(u=1+xy\). Alpöge's map is \(F=(u^3z+y^2u(4+3xy),\; y+3xu^2z+3xy^2(4+3xy),\; 2x-3x^2y-x^3z)\).
An independent certificate found in-session before the announcement was shown: \(F(1,-2,9)=F(-\tfrac13,4,27)=F(-\tfrac23,-\tfrac12,-\tfrac98)=(-1,1,-1)\). Exact rational arithmetic throughout; the determinant identity checked by two independent implementations.
\(F\) is equivariant for \(\lambda\cdot(x,y,z)=(\lambda^{-1}x,\lambda y,\lambda^2z)\), invariants \(t=xy\), \(w=x^2z\). With \(m=F_3/x\), \(\tilde g=xF_2\), \(\tilde h=x^2F_1\) (at \(x=1\)), the map descends to \(G=(m\tilde g,\,m^2\tilde h)\), the \(x\)-direction multiplied by \(m\).
Proof sketch. Chain rule through \((x,y,z)\mapsto(x,t,w)\) gives \(\det JF=\operatorname{jac}G/m^2\); expand \(\operatorname{jac}(m\tilde g,m^2\tilde h)\). ∎
Consequences: all shadow ramification lies on the single crushed line \(\{m=0\}\) with multiplicity two, at every degree — higher degree is more sheets over one crease, never more creases. Alpöge satisfies the equation with \(c=2\); Gallagher's degree-4 map with \(c=-1\); the Zyskind–Sol factory in generalized weights \(A=1+xy^m\). Gallagher's seed conditions (\(p(0)=0\), \(p(1)=-c\), \(\int_0^1p=0\); fiber degree \(\deg p+1\)) say the potential \(P=\int p\) returns to zero: counterexamples correspond to zero-net-work cycles.
Proof. No degree 2: an index-2 extension is Galois, and a Keller map with Galois extension is an automorphism (Campbell 1973). All \(n\ge3\): Gallagher's seed family (degree 4 verified exactly, [C9]); degree 3 is Alpöge's map; composition closes the monoid (degrees multiply; multiplier cocycle \(m_{12}=m_2\cdot(m_1\circ G_2)\), verified [C8]). ∎
So 3 is the least possible counterexample degree — the first non-Galois opportunity, and it was taken.
With Moh (\(\deg\le100\)) and the prime-extension exclusion (arXiv:2407.13795), three partial walls; none decides \(\mathrm{JC}_2\). Classically, a proper étale self-map of \(\mathbb{C}^n\) is a trivial covering: in a closed universe the conjecture is true; every counterexample is an open-system phenomenon. On the honest torus the phenomenon is trivial (\(x\mapsto x^2\), log-Jacobian 2).
At \(F\) (deformations of degree \(\le7\), two primes): tangent space dimension 33; joint composition moves (with cross-cancellation) plus equivariant deformations explain 32; the residual direction, pure torus-weight offset \(+3\), is obstructed at second order, cross-terms included. At Gallagher's \(F_4\): \(\dim T=11\), fully explained after the deep conjugation audit. Within stated caps, the torus symmetry is locally forced at both founding examples. Caveats: degree caps, two basepoints, modular arithmetic (two primes), local only.
verifier.py: nine checks (C1–C9),
exact symbolic arithmetic, independent determinant reimplementation, exit
nonzero on failure.
JacobianTorus.lean: Lean 4 /
mathlib draft — torus triviality, collision certificates, determinant
as ring identity, dim-2 wall; compile status published, including the
sorry count.
Q1 (generation): is every étale self-map of \(\mathbb{C}^3\) a composition of automorphisms and quasi-torus lifts? Q2 (equivariance forced): do non-quasi-torus counterexamples exist in any dimension? Q3 (quantization): exhibit the explicit non-surjective endomorphism of \(A_3\); compare index with fiber degree. Q4 (the plane): \(\mathrm{JC}_2\), now with a structural target: rule the non-toric mechanisms in or out.
Attribution & timeline. Alpöge (counterexample, 07-19); Gallagher (seed family, 07-20); Zyskind–Sol (factory, 07-20); Zhang (consequences, circulating); this project (master equation, spectrum lower half, monoid/cocycle, quasi-torus wall, rigidity, verification suite). Corrections recorded, never erased: the dustbin. Ledger: CLAIMS.md.