Nonvacuity and harmonic restriction

For the example plane, choose ${\mathbf{x}}_{\ast }$ with $a_{ij}({\mathbf{x}}_{\ast })\ne 0$. Choose compact scalar position data with Hessian ${\partial }_i{\partial }_j\varphi (t_0,{\mathbf{x}}_{\ast })=a_{ij}({\mathbf{x}}_{\ast })$ and zero scalar velocity. A cutoff times a local quadratic polynomial gives such data. Then

Equation (eq:Duhamel) gives

A nonnegative smearing in a small future region gives $B_f(e,\varphi )\ne 0$. Thus the restricted construction need not represent the zero functional.

If a harmonic prescription has compensator $w={\mathsf{G}}_{1,R}D(k)$, or the zero-data solution of $\square w=D(k)$, then on the plane $D(k)=0$ and $w=0$. Its restricted pairing equals the raw pairing. This requires only the prescribed zero solution for a zero source; it does not extend a retarded inverse to an undeclared source class.

The compact integral $F_{J_c}(k)$ is meaningful on the Fourier-gap plane, but that does not identify its non-spacelike- compact waves with the phase space of an FH algebra. Nor do two interpolation conditions imply a dense-TT theorem.