The MSJS Paradigm: A Universal Finite-State Abstraction for Discrete Dynamical Systems and Cryptographic Applications

I present a fundamental paradigm shift in the analysis of discrete dynamical systems, formalized through the MSJS (Modular Special Jump System) framework.I demonstrate that the apparent complexity of iterative maps, exemplified by theCollatz conjecture, is an artifact of scale, not structure. By rigorously decouplingthe ”pattern” (the finite-state core) from the ”magnitude” (the scaling factor), Iprove that the dynamics are governed entirely by a closed, finite-state transducer.This leads to a universal criterion: any discrete system whose lower-order evolution is independent of its higher-order terms is inherently reducible to a finiteautomaton. Consequently, the problem of infinite convergence is transformed intoa finite combinatorial verification. I extend this principle beyond number theory,establishing a novel framework for Algorithmic Design and Cryptographic One-WayFunctions, leveraging the inherent asymmetry between forward computational easeand backward state reconstruction.This is a preprint version. The proof is presented in full and is open for peer reviewand discussion