A compact list of proved dependencies

The main result uses only the following ingredients:

1. the explicit action (eq:switchedaction);

2. coefficientwise Cauchy existence and uniqueness;

3. the support condition $\overline{U}\subset P_g$;

4. the naturality of the unswitched scalar density equation;

5. formal inversion of a map with leading term $\mathrm{id}$;

6. the full second-order metric pullback (eq:hprime)–(eq:kprime).

No quantum product, counterterm classification, earlier-series uniqueness theorem, or finite response matrix is used.

The Einstein–scalar restriction additionally uses the explicit coefficient definition in Appendix app:curvature and the naturality of the Einstein tensor. It is not needed for the off-shell classical intertwiner.