Prescribed extension spectra of mock automorphisms over finite fields

We classify the finite-extension permutation spectra occurring in an explicit family of mock automorphisms over finite fields. Every finite nonempty divisor-closed subset of $\N_{>0}$ occurs; in particular, this disproves Maubach and Willems conjecture that every mock automorphism permutes infinitely many finite extensions.
Every finite-spectrum map satisfying \[
\Jac(F)=I_n,\qquad F|_{\F_q^n}=\id \] has total degree at least $p(q+1)$. For every $n\geq2$, this bound is attained by a map with spectrum $\{1\}$. In dimension one, the same bound and exact spectrum $\{1\}$ are attained over every odd field and over $\F_2$; for even $q>2$ we give a nonexceptional construction with an effective finite bound for its spectrum. If $\delta_p$ denotes the least degree of a nonlinear reduced permutation polynomial over $\mathbb F_p$, then the least degree $\mu_p$ of a nonexceptional one-variable mock automorphism satisfies \[ \mu_2=6,\qquad \mu_3=12,\qquad \mu_p=p\delta_p\quad(p\ge5). \] We also record a cofinite-spectrum criterion and isolate the surviving prime-to-$p$ question. Every nontrivial counterexample constructed here has geometric generic degree divisible by $p$.