Lemma: A smooth surface coefficient cannot be cancelled

Suppose, near a point of $\Sigma$,

where $A$ is locally bounded and $D$ is smooth on $\Sigma$. If $D$ is nonzero at that point, the point belongs to $\mathrm{supp}q$.

Proof. In any sufficiently small spatial neighbourhood choose a compact smooth tensor test $\zeta$ such that $⟨D,\zeta ⟩\ne 0$. Let $\rho$ be a compact smooth temporal function with $\rho (0)=1$, and test against

The surface pairing is the fixed nonzero number $⟨D,\zeta ⟩$. The bulk pairing is $O(\epsilon )$ by local boundedness. Their sum is nonzero for sufficiently small $\epsilon$. Its test support lies in the prescribed neighbourhood.