Proposition: Exact compact scalar test

Put $u_f={G_m}^{\mathrm{A}}f$ and

Then $q_{f,h,\chi }$ is real, smooth, and compactly supported, with

For every smooth free scalar solution,

The dependence support in the classical $h,{\varphi }_0$ variables is contained in

which is compact.

Proof. Transposition of the retarded Green operator gives

The switched coefficient is compactly supported, so the last integration by parts has no Cauchy-surface contribution. The other term obeys

This proves the formula and its sign.

The advanced support property bounds the first term by $L\cap J_{\eta }^{-}(K)$; the second is supported in $K$. Both sets are separated from $t=0$. Although $w[h]$ is defined here for $t\ge 0$, its product with $f$ extends smoothly by zero near and below $t=0$.

Finite propagation for the scalar response and coordinate Cauchy problem gives dependence only in ${\mathcal{D}}_K$. A compact observation set has bounded spatial causal past on a finite closed time slab, so ${\mathcal{D}}_K$ is compact.