Inertia from the Full Response Gram Matrix: Spectral Branches, the Critical Wall, and the Positive Cost of Elimination

A spectral threshold may indicate stability without revealing whether a system is reversible—or how its internal circulation affects the response to external forcing. This distinction raises a fundamental question: how can spectral stability, reciprocity, and irreversible transport be described within a common mathematical framework without losing the information that distinguishes them? This work develops an operator-theoretic bridge between Perron–Frobenius geometry and Hermitian response geometry. It connects weighted spectral analysis, nonreciprocal linear response, stationary probability currents, and the algebraic structure of coupled systems. The results distinguish information captured by scalar spectral indicators from information retained by the full operator response, including the orientation and strength of irreversible circulation. The framework encompasses finite-dimensional transfer operators, discrete- and continuous-time Markov processes, and extensions to Hilbert-space operators. It establishes exact relationships between several established mathematical descriptions while clarifying their distinct physical interpretations. The central message is that apparently equivalent spectral descriptions can conceal substantially different irreversible behavior. Recovering the full response requires preserving structural information that scalar diagnostics alone cannot provide.