B1: the interaction functional and the printed order

The proposed expansion is printed once:

We match $\lambda$ to one additional $\kappa$ power: since $\mathcal{O}$ is the displayed ${\kappa }^2$ coefficient, its selected response would contribute at ${\kappa }^3$ when $\lambda =\kappa$. This does not assert completeness of the full ${\kappa }^3$ expansion.

Both entries have field degree three. A single cross-contraction has degree four and a double cross-contraction has degree two. The latter is the requested coefficient linear in $\hslash$ after the retarded division by $\hslash$. Triple contractions lie beyond the order in (eq:order). No $R_2$ is introduced.

The Euler operator $V_{1,g}$ appears only in derivative calculations. It is not the first argument of a time-ordered product. Similarly, contractions involving metric derivatives of $\mathcal{O}$ are contractions of the admitted composite. They do not define a separate quantum product with ${\mathcal{J}}_{\mathrm{D}}$ as an argument.