Theorem: Robust detected class

Set

It is a closed real linear subspace, and

If one detector is nonzero on this stage, the detected class is open and dense there. Each detected observation remains detected, by the same witness, under sufficiently small stage perturbations.

Proof. Continuity gives closed kernels. A symbol is nonzero exactly when it is nonzero at some $(\mathbf{n},\mathbf{p})$, proving the complement identity. If a detector is nonzero, the common kernel is proper. A proper linear subspace of a topological vector space has empty interior; otherwise scaling an interior neighborhood would fill the space. Its closed complement is consequently dense and open. Continuity preserves a fixed nonzero detector value in a neighborhood of its witness.