Multiparameter Geometry of Reduced Response

A one-parameter response law describes how an effective system changes along a path, but it does not determine what occurs when different parameter directions are compared. In a genuinely multiparameter reduction, path dependence, orientation, and geometric compatibility become independent pieces of information. This work develops a multiparameter geometric description of reduced response for non-selfadjoint operator families. Left and right elimination generate two evolving graph geometries, and their interaction governs the covariant variation of the effective response. This makes it possible to distinguish changes in magnitude from changes carrying directional or geometric information. The framework connects local parameter dependence with curvature, holonomy, and mixed directional correlations. Global behavior along closed paths is compared with local geometric structure, while stability is studied near the loss of hidden-sector solvability. The main conclusion is that multiparameter reduction contains genuinely surface-level information that cannot be reconstructed from one-dimensional variations or diagonal response magnitudes alone. Mixed directional and left–right correlations are required to distinguish reduced systems that may look identical under purely local scalar diagnostics.