The conditional identity has an existence hypothesis

This is the conditional logic of X-G XG (:conditional), not an existence proof for $\ell$. It does not identify the conditional expression with the Bogoliubov image of an unchanged source-switched interaction.

General perturbative gravity and locally covariant BV constructions require their own domains, renormalization conditions, and comparison maps BFR. They are not automatic completion theorems for (eq:mixed) and its nonlocal insertion.