Abstract

We identify the classical equation spaces and gauge actions on which specified first-order relational scalar functionals descend. An exhibited Poisson-dressed symbol is not constant on the unchanged free product of linearized gravity with a massive scalar; the same two-representative gap survives spatial Fourier regularization. Moving the scalar does not repair that product: $X_{\xi }\varphi$ preserves every compact-data massive free solution only when $\xi$ is Killing. Retaining the forced correction $P{\varphi }_1=-L_h{\varphi }_0$ changes the equation space and supplies a genuine coefficient gauge action. A past-trivial retarded harmonic section then makes the complete first-order functional a quotient observable. For a triangular system with a fixed free scalar background, compact constrained de Donder Cauchy data modulo residual wave gauges form a Hausdorff, well-posed quotient. We construct a compact coupled Ward pair with strictly positive on-shell value, although the immediate adjoint-exact compact repair vanishes on shell. Poisson compensation, compensator choice, and canonical normalization are treated separately. Finally, the sourced successor is divided into invariant equations, a gauge-fixed forced response, and descent on a named quotient. None of those sourced constructions is completed here.