Corollary: Open dense failure in a specified test space

Fix the scalar and vector wave of Theorem thm:residualwitness. Let $Q⋐(0,T)\times {\mathbb{R}}^3$ have interior meeting an open region where $w={\xi }^1{\partial }_1\varphi >0$. In the Fréchet space

the failures of raw representability on $\mathcal{S}$, and of harmonic representability on the domains of Theorem thm:harmonic, each form an open dense set. Their common failure set contains

which is itself open and dense.

Proof. The last set is the complement of the kernel of a nonzero continuous linear functional. Such a kernel is closed with empty interior.

For the exact failure sets, the map $f\mapsto \mathbf{H}$ is continuous and linear into compact distributions with common support, and the map from $f$ to the two data in (eq:zetadata) is continuous and linear into compact smooth data with common support. Their zero sets are closed linear subspaces. The witness makes each proper. The classification theorems identify their complements with the claimed failure sets.