Lemma: Detector continuity

For fixed $g,{\varphi }_0,\mathbf{n},\mathbf{p}$, the map

is continuous real-linear on every ${C_K}^{\infty }(M;\mathbb{R})$ with $K\subset {M_T}^{\circ }$ compact.

Proof. The map $f\mapsto f{\partial }_{\mu }{\varphi }_0$ is continuous on the fixed test stage. Convolution with the advanced wave Green operator and differentiation produce a smooth function continuously. Multiplication by the fixed ${\sigma }_{\epsilon }$ localizes the source to $J^{-}(K)\cap \{t\ge 0\}$, a fixed compact set. A further advanced propagation, restriction to $t=0$, and contraction with $\mathcal{P}$ are continuous smooth operations. Finite propagation bounds the spatial support of that restriction by a fixed compact set. Its Fourier evaluation at fixed $\mathbf{p}$ is therefore a continuous smooth-function pairing. These are exactly the operations in (eq:detector).