Adaptive Lag and Transition Susceptibility (v3.1): Nonlinear State-Dependent Gain and Saddle-Node Threshold in Constrained Control Systems

This release (v3.1) presents a nonlinear extension of the Adaptive Lag framework for analyzing systemic vulnerability in constrained control systems. The model formalizes adaptive lag as a state-dependent tracking delay arising from bounded feedback permeability and structural rigidity. The internal configuration variable x(t) tracks a non-stationary environmental process E(t). Permeability is defined as a nonlinear function of tracking error, Phi(e) = Phi0 * exp(−gamma * e^2), yielding a state-dependent adaptive gain k(e) = Phi(e) / (R + epsilon). Under constant environmental drift, the resulting error dynamics admit an analytically derived saddle-node bifurcation. In dimensionless form, the critical threshold is nu_c = (1 / sqrt(2)) * exp(−1/2) ≈ 0.4289. Below this threshold, stable tracking equilibria exist; above it, fixed points annihilate and misalignment diverges. Deterministic phase mapping and stochastic simulations (Ornstein–Uhlenbeck forcing) numerically confirm the analytical boundary. A revised hazard formulation incorporates state-dependent noise amplification, distinguishing constructive from destabilizing variance regimes. This framework does not claim universal collapse prediction. Instead, it provides a minimal nonlinear control model for quantifying adaptive lag and transition susceptibility under bounded processing capacity. The release includes analytical derivations, numerical validation, sensitivity analysis, and reproducible pseudocode. Nonlinear Dynamical Systems Adaptive Control Theory Saddle-Node Bifurcation Stochastic Differential Equations Ornstein–Uhlenbeck Process Complex Adaptive Systems Feedback Saturation State-Dependent Gain Phase Transition Dynamics Systemic Fragility Transition Susceptibility Hazard Modeling Bifurcation Analysis Stability Thresholds Control under Environmental Drift