The scalar-only domain actually used

The scalar support is contained in $K$, whereas background dependence may extend throughout ${\mathcal{D}}_f$. Neither inclusion is a claim of equality for every configuration.

The retarded construction uses only:

1. A quadratic insertion $V=-\frac{1}{2}⟨\phi ,\mathsf{L}\phi ⟩$, where $\mathsf{L}$ is formally self-adjoint with smooth compact coefficients.

2. A specified linear representative $F_c$, with $c\in \mathcal{D}(M)$.

Joint admission is additional information, not a substitute for this named quantum domain.