Trajectory Break Principle (TBP): Universal Limits of Controllability in Classical, Stochastic, and Quantum Systems

The Trajectory Break Principle (TBP) establishes a universal limit on controllability in classical, stochastic, and quantum systems. Finite control capacity and finite relaxation time inevitably induce control‑driven trajectory singularities. By identifying two fundamental constants, Kmin⁡ and Kmax⁡, TBP defines a universal control zone that constrains all physically realizable trajectories. The principle unifies concepts from dynamical systems, information theory, and quantum mechanics, offering a quantitative framework for predicting when trajectories become unstable or uncontrollable. TBP provides a general, experimentally testable boundary for the applicability of control in physical systems.