Perturbative and quantum bookkeeping

The parameter $\lambda$ counts field perturbations:

It is not $\hslash$, a smoothing length, or an interaction coupling.

The concrete classical target is the complete coefficient through ${\lambda }^2$. The first prospective quantum calculation is the order-$\hslash$ contraction associated with quantizing its quadratic part in a chosen free representation. Section sec:quantum derives this contraction count.

An interacting order-$\hslash$ observable can require additional action vertices and observable coefficients, depending on the coupling grading. No complete interacting inventory is claimed from the free quadratic count.