The two-vertex products are prerequisites

The term

involves a construction with two interaction insertions and one observable insertion. In a time-ordered formulation its definition requires, among other ingredients, products containing $T_2(V_1,V_1)$ and the relevant three-insertion time-ordered product.

The term $R_1(V_1;{{\mathcal{O}}_f}^{(1)})$ creates a different problem: ${{\mathcal{O}}_f}^{(1)}$ contains the nonlocal coordinate kernel $G_{\square ,0}D(h)$. The domain and extension analysis for that kernel cannot be inferred from a calculation with an external smooth metric.

The extension theory of BF supplies criteria for numerical distributions once those distributions and their scaling degrees have been determined. It does not identify them for a new composite by notation alone.