The Perron--Hermitian Bridge for Nonreversible Response: The Circulation Gram Matrix and an Exact Resolvent Law

Is inertia necessarily a fundamental property of a physical system, or can its effective form emerge from the internal response of coupled degrees of freedom? This question becomes particularly significant when observable motion interacts with hidden sectors, potentially altering both the system's apparent inertia and its stability. This work develops a unified operator-theoretic framework connecting spectral curvature, complex response geometry, effective inertia, and hidden-sector elimination. It investigates how interactions modify the curvature of a spectral branch, under what conditions positive and finite inertia is preserved, and how critical behavior emerges when those conditions cease to hold. A central distinction concerns the information retained by different reduction procedures. The analysis identifies why seemingly equivalent descriptions can produce different effective responses and why local stability conditions may fail to capture global behavior. The framework also examines the role of complex correlations, singular limits, and the consistency of successive reductions. The results extend across finite-dimensional spectral systems, selected Hilbert-space settings, and convergent families of regulated models. By establishing exact connections between several established mathematical formulations, the work reveals a common structural mechanism underlying effective inertia, spectral stability, and the response of coupled systems.