Field content, kinetic operator, switched source, response

Two prescriptions are part of the definition. First, the background-field prescription: the terms of the fluctuation equation that do not depend on $u$, the Einstein tensor density of the off-shell background, are dropped (Lemma lem:fluct, Remark rem:tadpole). Second, the switch: the order-$\kappa$ gravitational and gauge-fixing sources are multiplied by X-D's ramp ${\sigma }_{\epsilon }$, so that $\mathsf{S}$ is past-compact and ${{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}\mathsf{S}$ is defined by the retarded propagator alone, in the same place where X-A's coordinate equation $\square {{\mathsf{X}}_1}^{\rho }=-{\partial }_{\mu }({\sigma }_{\epsilon }{a}^{\mu \rho })$ multiplies its source. The switch acts on the source, not on a density; the two differ by terms supported in the layer $0<t<\epsilon$, which are not used.