Affine monodromy and exact value distributions over finite fields

We study finite field value distributions through the fixed-point statistics of monodromy groups. For a regular degree-$n$ cover, omitted values are controlled by derangements. Thus natural symmetric monodromy gives support density $1-D_n\mathrm{/}n!\to 1-e^{-1}$, while the Cameron–Cohen bound gives the universal ceiling $1-1\mathrm{/}n$, attained by sharply $2$-transitive affine monodromy. We give an explicit polynomial realization of this optimal mechanism. For $N=p^e$ and $h\mid N-1$, set $${\mathrm{\Lambda }}_{N,h}(U)=U(U^{(N-1)\mathrm{/}h}-1{)}^h\mathrm{.}$$ Its geometric Galois closure is rational, $$U=z^h,T=(z^N-z{)}^h,$$ and its geometric monodromy is the affine group $({\text{F}}_N,+)⋊H_h$. For every extension ${\text{F}}_Q\mathrm{/}{\text{F}}_p$ we compute the complete fibre enumerator of ${\mathrm{\Lambda }}_{N,h}$ exactly, including the nonregular cases. In the full affine case $h=N-1$, the polynomial $$U(U-1{)}^{N-1}$$ attains the Wan–Shiue–Chen upper bound for non-permutation polynomials over every finite field containing ${\text{F}}_N$; over arbitrary extensions we compute the exact defect from that bound.