A well-defined boundary charge

At a fixed configuration $\Phi$, a metric derivative of an element of ${\mathcal{P}}_{\Sigma ,\min }$ has the form

where $u_{\Phi }$ is a compact distribution with no timelike wavefront covectors and $r_{\Phi },v_{\Phi }$ are smooth near $\Sigma$. The boundary coefficient $v_{\Phi }$ is spatially compact.