Operator-Spectral Theory of Institutions: Norm Coherence, Trace-Class Obstruction, and Empirical Validation

This manuscript constructs a unified operator-theoretic framework for modeling the stability and collapse of institutions—ranging from educational systems and corporations to street organizations and high-opacity ("secret") societies. By representing institutions as dynamical objects on a Hilbert space, the paper introduces a "coherence Laplacian" whose ground-state spectral gap strictly quantifies the consistency between institutional norms and membership actions. The framework proves that institutional stability is completely governed by this spectral gap. A robust institution maintains a strictly positive gap, while institutional collapse is mathematically equivalent to the gap vanishing. By isolating these structural invariants from subjective objective functions, the theory provides a universal, cross-domain baseline for institutional analysis. To formalize this collapse transition, the manuscript utilizes advanced spectral geometry, identifying trace-class obstructions via regularized Fredholm determinants and the Birman-Schwinger principle. It further extends into Zeta-regularized spectral action, demonstrating how odd-zeta sectors capture nonlinear, higher-order structural feedback (e.g., triadic norm-action-reputation frustration). Alongside the analytic theory, the paper provides a reproducible, finite-dimensional JAX/Python numerical simulation. This empirical surrogate models norm-to-member coupling, boundary enforcement, reputation, and opacity, demonstrating a sharp, determinant-based phase transition. The numerical results confirm the theoretical bounds, showing exactly how increasing institutional opacity forces a spectral rupture, drives the regularized determinant to zero, and triggers systemic collapse.