The general variational background gate

For a new density-switched variational sheet, a more general proof must construct:

1. retarded products on all actual local vertices and the nonlocal relational insertion, including every coefficient in (eq:fullcoefficient);

2. quantum comparison maps for the changed background splittings, with the required perturbative agreement;

3. transport of causal and Hadamard data, rather than an unproved free pullback automorphism;

4. equation-ideal compatibility and the relevant finite Ward normalization.

If these data produce the finite intertwiner

then extraction of its ${\kappa }^2\hslash$ coefficient gives the requested background identity. Coefficient extraction is immediate; construction of the intertwiner is not.

The distinction is precisely why general local covariance and the classical structural argument cannot substitute for the missing quantum operation HW,TTZ.