Assumption: VII-C's Normal-topology Wick extension, copied

A proposed extension is a Hausdorff locally convex $\mathbb{C}[[\hslash ]]$-algebra $\mathcal{A}$ such that:

1. ${\mathcal{P}}_{\mathrm{mc}}[[\hslash ]]$ embeds into $\mathcal{A}$, with the fixed Wick product.

2. $\mathcal{A}$ admits the sharp insertions $Q_{\Sigma }(B_f)$ with $[Q_{\Sigma }(B_f){]}_{{\hslash }^0}=B_f$ on all smooth configurations.

3. On the degree-$(1,1)$ kernel stages, admission is sequentially continuous for the normal topology and agrees with the old inclusion on regular kernels; in particular $B_{f,\epsilon }\to Q_{\Sigma }(B_f)$ in $\mathcal{A}$.

4. Multiplication is hypocontinuous for bounded sets, and the field-independent ${\hslash }^3$ coefficient map $a\mapsto {\mathrm{ev}}_0[{\hslash }^3]a$ is continuous and agrees with the usual extraction on the old domain.