Switching, free-null tests, and initial data

On a free scalar,

All derivatives of $g$ are retained. The equality

holds when $t_0$ precedes the compact vertex support. It does not identify that expression with the unswitched zero-data response $S_{t_0}(h)\varphi$.

The witness $f=P\rho$ has

The retarded insertion is nevertheless

because its off-shell input label was retained before representation. The fixed contact $P{W}_m^{\mathrm{F}}=i\delta$ is essential to this conclusion.

For a pure gauge,

The homogeneous subtraction is the initial-data contribution. It is absent only when those data vanish. A general gauge transformation of the triangular scalar response does not preserve its zero-data condition.