Five corrections that determine the scope

The following distinctions are needed before formulating the theorems.

1. The pair $(\eta ,0)$, with zero cosmological constant, is an exact Einstein–scalar background solution. A nonzero free scalar on a fixed Minkowski metric is generally not an exact background solution at nonzero gravitational coupling.

2. The bare dressing coefficient is not the first interacting coefficient. If both constructions exist, the first interacting coefficient is

   $${{\mathcal{O}}_f}^{(1)}+R_1(V_1;{{\mathcal{O}}_f}^{(0)}).$$

   An external-metric scalar response supplies only the retarded term.

3. Turning on a second-order source does not retrospectively alter a previously fixed first-order coefficient. It changes the equations and data needed at the next order.

4. A second-order interacting coefficient requires products involving two first-order vertices. Consequently, the required $T_2(V_1,V_1)$ and three-insertion constructions cannot all be postponed until after that coefficient has supposedly been proved.

5. A fit on finitely many smearings does not prove a global counterterm identity. Conversely, a separator is valid only against the complete allowed response space.