Lemma: Nonzero sharp charge for every nonzero test

For every $0\ne f\in \mathcal{T}$, there is a real smooth scalar configuration $\varphi$, compactly supported in $V$, with

Proof. The metric derivative is $q_{a_{f,\varphi }}$, with

Choose $x_0=(t_0,{\mathbf{x}}_0)$ with $f(x_0)\ne 0$. Choose an initial point $\mathbf{z}$ satisfying $|{\mathbf{x}}_0-\mathbf{z}|=t_0$. It lies in the retained initial ball. On a small neighbourhood $O⋐V$ of $x_0$, $r(\mathbf{x})=|\mathbf{x}-\mathbf{z}|$ is smooth and positive. Choose a nonnegative real $\chi \in \mathcal{D}(O)$, positive at $x_0$, and set

On the cone $t=r(\mathbf{x})$, ${\partial }_t{\varphi }_{\mathbb{C}}=\overline{f}\chi$. Equation (eq:advancedtrace) gives

Write ${\varphi }_{\mathbb{C}}={\varphi }_1+i{\varphi }_2$ with real compact smooth parts. Complex linearity implies that at least one part has nonzero trace. Choose it.