A Cross-Domain AD/ZF Framework for Rupture and Critical Transitions in Complex Systems

This work introduces a cross-domain, testable, and fully reproducible mathematical framework for rupture in complex systems based on the interaction between structural constraints (ZF) and adaptive stability (AD). Structure is modeled as a potential function, while adaptation is defined as the local curvature of this potential, interpreted as a stability operator. A dimensionless tension variable T=∣F∣/∣AD∣ predicts rupture when external forcing exceeds adaptive stability. The framework is demonstrated using a nonlinear dynamical system applicable to biological regulation, cognition, and AI overload. Simulations reproduce classical early-warning indicators such as variance increase and critical slowing down. Rupture is interpreted as a loss of stability of a Lyapunov basin structure. The full implementation is reproducible and included in the appendix.