Definition: Completed named sector

Let ${\mathcal{W}}_H$ be the algebra of finite ${\star }_H$-words in compact linear fields and

with coefficients in $\mathbb{C}[\hslash ,\kappa ]$. Set $\mathfrak{m}=(\hslash ,\kappa )$. The sector ${\mathcal{S}}_H$ consists of $X\in {\mathcal{E}}_H$ such that, for every $p\ge 1$, some $W_p\in {\mathcal{W}}_H$ satisfies $X-W_p\in {\mathfrak{m}}^p{\mathcal{E}}_H$.

This is the formal coefficientwise completion, not closure in the analytic topology introduced later. For admissible $H^{\prime }$, define safely ${\mathcal{S}}_{H^{\prime }}={\bar{\alpha }}_d({\mathcal{S}}_H)$ once the ordering map is established.