Definition: Exact source-preserving variational upgrade

An upgrade in the class considered here must satisfy all of the following:

1. the free metric operator is ${\mathsf{P}}_{\mathrm{gr}}$, with the retarded inverse ${{\Delta }_{\mathrm{gr}}}^{\mathrm{R}}$ and its frozen causal relations;

2. the interaction is even and has a symmetric bosonic second variation after any algebraic auxiliary elimination;

3. for a compact background $w$, its reduced mixed metric Hessian $L_w$ has compact coefficients, or more generally a compact two-point kernel, so its bilinear pairing on smooth free inputs is defined and symmetric;

4. its classical retarded mixed field response obeys

   $$-{{\Delta }_{\mathrm{gr}}}^{\mathrm{R}}{L}_{w}u=-{{\Delta }_{\mathrm{gr}}}^{\mathrm{R}}{N}_{w}u$$

   for every smooth free metric solution ${\mathsf{P}}_{\mathrm{gr}}u=0$ and every allowed smooth background, including compact backgrounds in the onset layer;

5. quantum corrections are formal series in nonnegative powers of $\hslash$, with the classical response in (U4) unchanged.

Equality in (U4) is equality of fields, equivalently equality after every compact smooth linear metric smearing. It is stronger than equality of one chosen relational observable.