Proposition: Finite diagnostics are not the full topology

No finite family of linear scalar diagnostics on ${\mathcal{P}}_{\Sigma }$ determines every admitted functional. In particular, the finite evaluation table does not decide all boundary traces, all wavefront conditions, or convergence of arbitrary families in ${\tau }_{\Sigma }$.

Proof. For $N$ diagnostics choose $N+1$ linearly independent regular scalar linear functionals with disjoint nonzero smearings in $U$. They belong to ${\mathcal{P}}_{\mathrm{reg}}\subset {\mathcal{P}}_{\Sigma }$. The map from their span to ${\mathbb{C}}^N$ has a nonzero kernel. Thus a nonzero admitted functional has all listed readings zero. A finite list cannot replace the defining family of topology coordinates. No universal decision procedure is introduced by storing finitely many values.