Abstract

A fixed-time Klein–Gordon response to a linearized vacuum metric, with the scalar held fixed, is reconstructed as a compact tensor distribution with an explicit initial-surface term. On the spacelike-compact vacuum solution space it has a smooth compact conserved representative if and only if its gauge defect vanishes. After a retarded or zero-Cauchy harmonic prescription, the criterion is vanishing of the propagated defect. A compact-Cauchy residual wave proves failure on an open dense family of smearings in a specified Fréchet test space. The same pairing, restricted to any declared Fourier-gap two-dimensional spatial-TT plane, nevertheless has an explicit compact conserved representative, constructed from curvature currents dual to that plane. Separately, the first retarded product of the compactly switched minimal Einstein–scalar vertex with one scalar insertion is computed for an external metric and for a one-oscillator metric. The external coefficient is a linearly smeared free field; its contraction coefficients have scaling degree at most three and unique minimal extensions. The oscillator coefficient reconstructs the switched bilinear and defines an essentially self-adjoint real polynomial on a common core. The Newton–wave composition of stress, inverse, and scalar response is treated as a distinct cubic Wick form. Dense-TT representability, the adiabatic limit, and higher causal products remain open.