Lemma: Local advanced trace formula

For a compact smooth source $a$ in $V$, its supplied chart advanced trace is

For source support in $V$, every relevant cone segment and initial endpoint lie in the retained laboratory.

Proof. The spatial transform of $\delta (s-|\mathbf{x}|)/(4\pi |\mathbf{x}|)$, $s>0$, is $\sin (s|p|)/|p|$. For $\square =-{{\partial }_t}^2+\Delta$, the zero-future-data mode equation has value

at $t=0$. Inverse spatial transformation gives (eq:advancedtrace). Saturation and $R>r_0+T_0$ give the support assertion. This calculation represents the prescribed local comparison response, not a global cone inverse.