The Residual after Elimination: From Orthogonal Innovation to Effective Response

We investigate the information retained after eliminating hidden degrees of freedom from a finite-dimensional positive-definite Hermitian operator. A single energy-dependent elimination signature connects orthogonal residual geometry, linear innovation, Feshbach–Schur reduction, resolvent response, and dynamical memory. Its value at zero determines the minimum energy after elimination, while its negative first derivative is the Gram matrix of the full-state reconstruction. The main result solves an extremal problem over all positive realizations sharing the same signature value, first derivative, and known lower spectral threshold of the hidden block. We construct a minimal canonical realization that saturates a matrix-valued lower bound throughout the subthreshold energy interval. We determine the exact minima and suprema of the first indexed eigenvalues, allowing arbitrary finite hidden dimensions. All lower minima are attained simultaneously by one realization, while a single family approaches all upper suprema. Operator equality with the extremal bound at one energy enforces the entire extremal signature. The deviation from this bound controls the high-energy component of the static reconstruction and satisfies optimal comparisons across energies. The full signature identifies the active hidden realization up to unitary equivalence, whereas first-order data determine only its minimum required dimension. Noncommuting and three-mode examples demonstrate the benefit of retaining complete matrix information and distinguish reduced visible-state weight from destructive interference in a fixed spectral readout.