Lemma: Transposition

For a compact interior test $a$ and smooth source $s$,

The formula also holds for smooth compact tests flat at $t=0$, extended by zero to negative times.

Proof. Use Green's identity with $u={\mathsf{E}}_{\mu }s$ and $b={\mathsf{E}}_{\mathrm{A},\mu }a$. The terminal term vanishes because $b$ is terminally zero; the initial term vanishes because $u$ has zero Cauchy data. Spatial boundary terms vanish after a cutoff outside the compact retained past. The flat-at-zero case follows by the same identity or smooth cutoff approximation.