Lemma: Compact detection

Let

be a nonzero compactly supported distribution whose support is contained in the interior of $M_{+}^2$. Suppose the configuration space permits arbitrary smooth scalar and metric configurations. Then ${\mathcal{C}}_{f,g}\ne 0$ as an actual functional.

Proof. There are compactly supported smooth functions $u(x),v(y)$ with

Indeed, finite sums of tensor-product test functions are dense in the test-function space on a product. This can be seen by a partition of unity into product charts followed by smooth Fourier approximation on compact rectangles. Continuity of a distribution would force it to vanish if it vanished on every such tensor.

Extend $u$ smoothly by zero where necessary and choose

Then ${\partial }_t\varphi =u$. Choose the smooth metric configuration with $h_{00}=v$ and all other components zero. Equation (eq:commutator) gives a nonzero value. If only real configurations are used, decomposition into real and imaginary parts supplies real $u,v$ detecting the complex distribution.