Proposition

Gate 1 is the mixed scalar Noether identity, not an autonomous master equation $(S_{\mathrm{sc}},S_{\mathrm{sc}})=0$. With coupling-stripped densities $S_{\varphi ,i}=\frac{1}{2}\int {a_i}^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi$ and $a_0=\eta$, ghost normalization $a=\frac{1}{2}$ distinct from $\alpha ,\beta$, and $sq=\kappa a{\mathcal{L}}_{c}q$, the densitized identities

integrate, under compact support, to

Three objects stay separate: the invariant baseline $S_{\mathrm{inv}}$; the prepared-action defect $(S_D,S_D)$; and the insertion-class obstruction of XVI-N, thm:massless-obstruction.