Lemma: Compact detection

If $fg(K-\overline{K})$ is a nonzero compact distribution in the interior observation chart, then ${\mathcal{C}}_{f,g}\ne 0$ as an actual functional.

Proof. Density of finite sums of product tests gives smooth compact $a(x),b(y)$ with a nonzero pairing. Choose a smooth scalar with

and choose $h_{00}=b$, all other components zero. These are allowed off-shell configurations. For real-only configurations, real and imaginary decomposition supplies a real detecting pair.