Spectral Dynamics of Intensity, Clarity, and Regulation: Operator-Theoretic Closure of Post-Volatility Behavioral Systems

This manuscript establishes a rigorous operator-theoretic and nonlinear dynamical framework for modeling post-volatility behavioral systems. It introduces a strict mathematical separation between "intensity" (behavioral amplitude) and "clarity" (temporal invariance), proving that true stability arises from anchor dominance over destabilizing external perturbations rather than resonant amplification. The system is modeled within a continuous state space encompassing self-weighting, validation dependence, empathy bandwidth, and regulation strength. By mapping these nonlinear dynamics onto a Fredholm decomposition, the manuscript demonstrates that systemic psychological or behavioral collapse can be exactingly modeled as a spectral rupture. Utilizing the Birman-Schwinger principle, it proves that structural instability occurs precisely when the regularized Hilbert-Schmidt determinant vanishes. To formalize higher-order stability, the framework incorporates a Heat Kernel and Zeta Expansion layer. This odd-zeta hierarchy extracts exact mathematical observables for low-order spectral strain, mid-order instability curvature, and deep ultraviolet regulation sensitivity. Finally, a reproducible JAX numerical simulation is provided, visualizing the exact moment of catastrophic spectral collapse—demonstrating how an increasing external validation load pushes the Birman-Schwinger minimum eigenvalue across the absolute failure threshold.