Field content before any gate

A, frozen here. The scalar is the free quantum field of $P_m=\square +m^2$ on Minkowski space, $m\ge 0$. The metric coefficients are smooth, classical, and off shell. There is no quantum metric propagator and no gravitational action.

The slab and initial surface belong to the classical ramp prescription. Its coordinate sources vanish for $t\le 0$, so the zero-data coordinate inverse agrees with the retarded inverse on those sources. No quantum Cauchy data are prescribed. The quantum field is defined on all of Minkowski space.

B, named and not opened. Quantizing $h$ would require a specified quadratic graviton action, gauge fixing, and a different contraction domain. None is present in the classical source manuscript. An additional graviton response would not follow from the scalar-only normalization below. Its Ward formulation and possible loop graphs would require separate analysis.

The present theorem makes no assertion that every such successor graph exists or fails. In particular, a schematic triangle is not a proved obstruction merely because it contains a partial diagonal. Branch B is a different construction, not a correction to a missing term inside Branch A.