Finite Orbit Theorem for a Binary XOR–Shift Dynamical System

We study a discrete dynamical system defined on nonnegative integers through binary operations involving XOR and a shift transformation. We prove that the induced map does not increase the binary length of the iterates, implying that the associated state space is finite. As a consequence, every orbit of the system is globally bounded and eventually periodic by a direct application of the pigeonhole principle. This result establishes a finite–orbit theorem for a class of binary XOR–shift dynamics and provides a simple structural criterion ensuring the absence of divergent trajectories.