Not constructed here

The following are named, not supplied:

1. the gravitational self-interaction tree associated with the cubic Einstein–Hilbert density;

2. the other tree terms required by a different background-gauge completion;

3. a complete second-order interacting quantum metric carrying the missing tensorial transformation;

4. the mixed scalar–graviton loop at ${\kappa }^2\hslash$;

5. a time-ordered hierarchy, normalization family, ghost sector, or loop subtraction.

If $S_3[r]$ denotes the cubic Einstein–Hilbert coefficient, the self-response has the schematic affine insertion

This term is of tree order, not excluded by a leg count. Its construction is not undertaken here.

Likewise the insertion ${\hat{A}}_1[\mathbf{r}]$ is already bilinear in free fields. A second cross-contraction with $V_{1,g}[\mathbf{r},\mathbf{\varphi }]$ can produce a loop coefficient. Tree exactness of an earlier scalar-linear response rule is not transferred to that mixed insertion. No claim of a vanishing positive-$\hslash$ remainder is made.