Proposition: Positive-width regular sector

Through coordinate order two, every coefficient of ${{\mathcal{O}}_{f,ϵ}}^{\mathrm{conv}}$, for fixed $ϵ>0$, has a smooth compact kernel. Consequently these coefficients belong to ${\mathcal{P}}_{\mathrm{reg},\mathrm{pol}}$ and have all finite free Wick products in the ordinary polynomial algebra.

On fixed smooth configurations the coefficients converge to their sharp classical counterparts as $ϵ\downarrow 0$. This is not convergence of their quantum products.

Proof. Each sharp coefficient is a continuous finite polynomial in smooth fields, with compact multilinear kernel by the kernel theorem and finite propagation. Convolution in every field slot makes that kernel smooth and enlarges support only by a fixed compact $ϵ$-neighbourhood.

On fixed smooth configurations, convolution converges in all smooth seminorms on the relevant causal compact. Continuous dependence of the coefficientwise Cauchy problems then proves the classical convergence.