Response Reduction in Hilbert Spaces: Hidden-Sector Geometry and Singular Regimes

Eliminating hidden degrees of freedom is often treated as a straightforward substitution problem. In infinite-dimensional systems, however, an effective response may remain well defined even when no hidden state realizes the corresponding extremal value, and several notions that coincide in finite dimensions can separate. This work develops an operator-theoretic framework for response reduction in Hilbert spaces that keeps these distinctions explicit. It studies hidden-sector elimination, singular limits, complex response correlations, and the relation between energetic and oriented reductions without assuming that conventional inverse or pseudoinverse formulas remain available. The framework also addresses approximation and regularization. Finite-dimensional truncations, singular hidden sectors, and successive reductions are compared at the operator level in order to determine which parts of the effective response remain stable and which require additional geometric information. The resulting picture separates effective value, attainability, solvability, orientation, and correlation structure. In infinite dimensions these are genuinely different layers of information, and a reduced description that preserves only one of them can miss features that survive in the full response.