This research introduces a discrete dynamical system that maps the trajectories of all positive integers onto a structured 2D coordinate grid. Utilizing the reduced Collatz graph as a functional baseline, we define a global state-space operator aᵢ (n) and a corresponding topological transport mechanism for all odd integers. By assigning a recursive temporal index i to every odd spatial value n, this index i serves as a monotonic coordinate for the discrete transport of state values between orbits. The resulting deterministic predecessor map reveals structural symmetries between reductive modular branches and restorative growth operators, providing a new methodological foundation for the investigation of the Global Attraction Conjecture across the entire set of positive integers via the Fundamental Theorem of Arithmetic and the principle of modus ponens. The symbolic operators defined herein establish a global 2D topology; visual representations of these coordinate mappings will be provided in subsequent expanded versions. This work provides a spatial coordinate solution to the topological transport problems described in Abascal, 2024. Author ContributionsDamian Hartman is the primary discoverer of the framework described herein. Both authors contributed to the verification and formalization of the findings. Authors are listed in order of contribution.
Hartman et al. (Tue,) studied this question.