Lemma: Compact primitive

If $I$ is a connected open interval and $T\in {\mathcal{E}}^{\prime }(I\times N)$ satisfies ${\partial }_{t}T=0$, then $T=0$.

Proof. A compact test of integral zero on $I$ has a compact primitive. Thus $T$ depends on a time test only through its integral. Choose an integral-one test disjoint from the compact time projection of $\mathrm{supp}T$. Its pairing is zero, so every such constant-in-time pairing is zero. Product-test density gives the result.