This paper presents the rigorous mathematical core of the Judgment Exposure Subsystem within the Cognitive Amplification Inequality Theory (CAIT) program. It formulates a coupled nonlinear state space model that captures the endogenous interaction between judgment load, psychological exposure, utilization quality, Human–AI coupling depth, and realized intelligence under bounded exogenous amplification and capability fields. Under mild regularity and positivity assumptions the paper proves existence and uniqueness of equilibria via a scalar consistency map, derives local stability conditions from the Jacobian spectrum, and establishes eigenvalue sensitivity showing exponential gain with respect to coupling depth. A Hopf bifurcation analysis yields an explicit approximation for the critical coupling threshold λc that separates stable amplification from oscillatory overload regimes; center manifold and normal form reductions clarify how inequality curvature shifts the effective bifurcation parameter. A Lyapunov candidate and forward invariant region provide constructive sufficient conditions for global boundedness. Representative numerical experiments illustrate the phase diagram (stable, oscillatory, collapse regimes) and validate analytical thresholds. Managerial implications are summarized succinctly: the results identify the mathematical levers (damping, utilization quality, paced coupling) that preserve realized intelligence while avoiding instability. Full proofs, extended derivations, and simulation code are provided in the appendices and companion supplement.
Usman Zafar (2026) studied this question.