Adaptive Lag and Transition Susceptibility (v3.1.1): Clarified Nonlinear State-Dependent Permeability and Saddle-Node Threshold in Constrained Control Systems

Version 3.1.1 provides formal clarifications to the nonlinear extension of the Adaptive Lag framework introduced in v3.1. The model replaces the previous linear tracking formulation with a state-dependent permeability function Phi(e) = Phi0 * exp(−gamma * e^2), yielding a bounded positive adaptive gain k(e) = Phi(e) / (R + epsilon), with R > 0 and epsilon > 0 ensuring strictly positive and finite gain under all admissible states. Under constant environmental drift and smooth permeability decay, the reduced one-dimensional tracking dynamics admit a saddle-node bifurcation. After nondimensionalization (gamma = 1), the analytically derived critical threshold is nu_c = (1 / sqrt(2)) * exp(−1/2) ≈ 0.4289. Deterministic phase diagrams and stochastic simulations (Ornstein–Uhlenbeck forcing) confirm the analytical boundary between stable tracking and divergence. A revised hazard formulation introduces state-dependent noise amplification, distinguishing constructive from destabilizing variance regimes. This framework does not claim universal collapse prediction. It formalizes a dynamical vulnerability mechanism under bounded feedback capacity in nonlinear control systems and complex adaptive systems. Community feedback, replication attempts, and critical evaluation are welcome. Nonlinear Dynamics Saddle-Node Bifurcation Complex Adaptive Systems Feedback Permeability State-Dependent Gain Adaptive Lag Control Theory Stochastic Forcing Ornstein–Uhlenbeck Process Systemic Fragility Transition Thresholds Feedback Saturation Dynamical Stability Hazard Modeling Constrained Control Systems