Finite-Rank RKHS Detector Completeness

We construct an explicit finite-rank reproducing-kernel detector for the model space $K_{B_P} = H^2/B_P H^2$ associated with a finite Blaschke packet $P = \{w_1,\dots,w_n\} \subset \mathbb{D}$. The principal observation is that the Gram matrix $G_{ij} = \langle k_{w_i}, k_{w_j}\rangle_{H^2}$ of the reproducing kernels at distinct packet points is a Cauchy-type matrix and is therefore nonsingular: in fact $\det G$ is, up to a positive multiplicative factor, the squared modulus of a generic Cauchy determinant $\prod_{i<j}|w_i - w_j|^2 / \prod_{i,j}(1 - w_i\overline{w_j})$.

Nondegeneracy of $G$ yields a $\operatorname{rank}$-$n$ family of evaluation functionals that separates every nonzero vector of $K_{B_P}$. We lift this finite-rank detector to a candidate inhabitant of the $\texttt{RKHSModelSpaceDetector}$ interface used by the Burnol/Blaschke no-phantom programme, and isolate the residual completion obstruction in a paired $\texttt{Vol6.Obstruction.FiniteRankDetectorObstruction}$ module. The argument is purely finite Hardy-space theory: it neither invokes Nyman density nor any equivalent of the Riemann hypothesis.