Lemma: Compact distributions have no constant primitive

Let $I=(0,\infty )$, let $N$ be a manifold, and let $T\in {\mathcal{E}}^{\prime }(I\times N)$. If ${\partial }_{t}T=0$, then $T=0$.

Proof. A distribution with zero derivative in a connected interval is constant in that variable. Here is the product-test argument. Choose $\chi \in {C_c}^{\infty }(I)$ with $\int \chi =1$. For every $\eta \in {C_c}^{\infty }(I)$, $\eta -(\int \eta )\chi$ has integral zero and is the derivative of a compactly supported smooth function in $I$. Hence

Choose $\eta$ with integral one and support in an interval disjoint from the time projection of $\mathrm{supp}T$. The left-hand side is zero, so every pairing on the right is zero. Product-test density gives $T=0$.