Coefficient-Level Böttcher Theory for Wild Superattracting Germs of Degree $p^e$

Let $p$ be an odd prime, let $e\ge 2$, and put $q=p^e$. We study the wild family $${\phi }_{r,e}(x)=x^q+q{p}^r{x}^{q+1}=x^{p^e}+p^{r+e}{x}^{p^e+1}(r\ge 0),$$ and the inverse Böttcher coordinate $f_{r,e}(x)=x{\sum }_{k\ge 0}{a}_k(r,e)x^k\mathrm{/}k!$ characterized by $${\phi }_{r,e}(f_{r,e}(x))=f_{r,e}(x^q)\mathrm{.}$$ For the clean family, we prove a complete mod-$p$ digit-sum law in the special fiber $r=0$. For the higher fibers $r\ge 1$, we prove a coefficient-level theorem consisting of a global digit-weight lower bound, a leading monomial theorem on divisible non-pure classes, a lag-$e$ pure-power recursion, and subadditivity of the induced digit weight. This yields the pure-power branch word $$(B^{e-1}A{)}^{\lceil r\mathrm{/}e\rceil }{B}^{\mathrm{\infty }}$$ and the radius formula $$\rho (f_{r,e})=p^{-{\theta }_{r,e}},{\theta }_{r,e}=p^{-e\lceil r\mathrm{/}e\rceil }(\frac{1}{p-1}+e\lceil r\mathrm{/}e\rceil -r)\mathrm{.}$$ We then prove a tail-stable extension. In the special fiber, $p$-divisible tails preserve the digit-sum law modulo $p$. In the higher fibers, tails satisfying $v_p({\vartheta }_h)\ge {\mathrm{\Lambda }}_{r,e}(h+1)+1$ lie beyond the clean-family initial ${\mathrm{\Lambda }}_{r,e}$-graded term and therefore preserve the leading terms, the pure-power branch word, the valuation asymptotic, and the radius. For $e=2$, this recovers the Salerno–Silverman degree-$p^2$ family and the Fu–Nie radius statement for the inverse coordinate in that family.