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

A proposed extension consists of a Hausdorff locally convex $\mathbb{C}[[\hslash ]]$-algebra $\mathcal{A}$, with the following properties.

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

2. The algebra admits the sharp insertions $Q_{\Sigma }(B_f)$, with

   $$[Q_{\Sigma }(B_f){]}_{{\hslash }^0}=B_f$$

   as functionals on all smooth configurations.

3. On the degree-$(1,1)$ kernel stages used above, admission is sequentially continuous for the normal topology and agrees with the old inclusion on regular kernels. In particular,

   $$B_{f,\epsilon }⟶Q_{\Sigma }(B_f)in\mathcal{A}.$$

4. Multiplication is hypocontinuous with respect to bounded sets. The field-independent ${\hslash }^3$ coefficient map

   $$a⟼{\mathrm{ev}}_0[{\hslash }^3]a$$

   is continuous and agrees with the usual coefficient extraction on the old polynomial domain.