Positive-characteristic sparse, sporadic, lacunary, and generic architecture for quotient maps

Let $p>0$, let $p\nmid d$, and let $h_d$ be the reduced separable quotient associated with $H_d^{\rm raw}$ in characteristic $p$. Assume the characteristic-zero case is settled and exclude the small overlaps $d=1,2,3$. For $a\geq1$, the relevant sparse branches occur when $d=p^a-1$ or $d=p^a+1$; the possible sporadic case is $(p,d)=(19,6)$; and the first nonsparse Frobenius–lacunary tower occurs when $p\ne2,3$ and $d=2p^a+1$. Writing $n=\deg h_d$ for the degree of the separable quotient, let $G_{h_d}$ denote its monodromy group. Positive-characteristic sparse/sporadic/lacunary/generic architecture. Apart from those stated sparse and sporadic alternatives, every remaining positive-characteristic case with $\deg h_d=n>1$ should satisfy

$$A_n\le G_{h_d}\le S_n.$$

More explicitly, the alternatives are: $h_d=\tau^2$ when $d=p^a-1$; the Kummer or Artin–Schreier Frobenius-sparse branch when $d=p^a+1$; the Klein-four sporadic quotient when $(p,d)=(19,6)$ under the certificate-execution hypothesis; and the first nonsparse Frobenius–lacunary tower when $p\ne2,3$ and $d=2p^a+1$. The paper proves primitivity of this tower for every $a$ and the displayed monodromy bounds for $a=1$, while the case $a>1$ is expected to follow from control of the explicit wild inertia representation. The parser reports that the sporadic alternative is forced by a computer-assisted theorem and that, without the certificate-execution hypothesis, the conjecture is false.