The Artin-Hasse $p$-section: weighted convolutions and $p$-adic recovery

Let $p$ be an odd prime, and let $a_n$ be the reduction modulo $p$ of the $n$th coefficient of the Artin-Hasse exponential. We study the weighted $p$-section convolutions $W_k$ formed from the coefficients $a_{kp}$. We prove the conjecture of Avitabile and Mattarei for $1<k<p$ and extend it to a single global power-series identity determining the entire sequence $(W_k{)}_{k\ge 0}$. This identity yields an explicit base-$p$ digit formula; in particular, $(W_k)$ is $p$-automatic and admits an explicit finite-state evaluator. Independently, the exact conjugated $p$-section equation gives a strict $p$-adic fixed-point iteration for the logarithmic derivative of the $p$-section. Big-Witt reconstruction then recovers the $p$-section, and hence all coefficients of the Artin-Hasse exponential, to arbitrary prescribed $p$-adic precision by a purely modular procedure.