Resolution of the Collatz Conjecture: Xiang's Model of Myriad Numbers Converging to One with Chain Cycles and the Foundations of Integer Dynamical Systems

The Collatz conjecture is completely resolved by constructing Xiang’s Model of Myriad Numbers Converging to One with Chain Cycles. This model is built on the set of positive integers and establishes a novel data structure and operational framework for infinite-state data spaces. Within the conceptual framework of dynamical systems, it creates a new set of algorithmic tools, providing a general solution for solving recursive problems with infinite-state spaces. On this basis, the theoretical system of integer dynamical systems is established, extending the current scope of dynamical systems theory. Integer dynamical systems are independent of classical discrete dynamical systems. Classical discrete dynamical systems typically assume that the state space is endowed with a topological or metric structure (such as real numbers, complex numbers, or manifolds), and the maps are usually continuous or differentiable. In contrast, the state space of integer dynamical systems is discrete in topology, representing a specialized breakthrough within arithmetic state spaces. Integer dynamical systems are also independent of general arithmetic dynamical systems. Arithmetic dynamical systems typically take the state space as points on algebraic varieties, with maps often being polynomials or rational functions. Integer dynamical systems restrict the state space to integer points, and the maps are often the restrictions of polynomials or rational functions from arithmetic dynamical systems to the integers. Moreover, integer dynamical systems include piecewise linear maps (such as the Collatz map), which fall outside the scope of classical arithmetic dynamical systems. Integer dynamical systems apply the methodology of computability theory, drawing an analogy with the current architectures of the main branches of dynamical systems, to establish a unified theoretical system of integer dynamical systems. The theoretical framework of integer dynamical systems has the potential to overcome bottlenecks in complex problems across multiple disciplines and to drive a paradigm shift in algorithmic approaches. Key application areas include pseudorandom number generators (PRNGs) and cryptography, complex system modeling and control, chaos versus noise discrimination, and AI interpretability and chaos prediction.