Distance-Dependent Trajectory Coupling and 2-Adic Polynomial Coalescence in the Collatz Graph

Prime Arithmetic and Geometric Structure in the Collatz Graph: Modular Coalescence, Trajectory Overlap, and Parity Dynamics The relationship between arithmetic structure and dynamical behavior in the Collatz graph raises questions that are not captured by conventional trajectory analysis alone. In particular, the behavior of nearby odd integers suggests the presence of systematic geometric patterns whose origin requires a more careful investigation. This work develops an algebraic and computational framework to examine trajectory convergence, modular structure, and parity dynamics within the Collatz system. Rather than treating numerical overlap as an isolated statistical observation, it investigates the underlying mechanisms through modular arithmetic, affine trajectory representations, and discrete dynamical analysis. The study combines exact algebraic identities with large-scale computational experiments to explore several interconnected aspects of the problem: Modular trajectory coalescence: an investigation of conditions under which distinct initial values enter identical Collatz trajectories, including the role of 2-adic residue classes. Arithmetic adjacency and trajectory geometry: a systematic examination of how initial distance relates to the structure and extent of trajectory overlap. Parity dynamics: an analytical treatment of parity transitions and their relationship with stopping-time behavior. Spectral structure: a probabilistic representation of parity sequences and their associated correlation and frequency-domain properties. Mersenne dynamics: an exact algebraic identity connecting the binary structure of Mersenne numbers with their Collatz evolution. Inverse-graph organization: a computational study of recurrent structures and dominant pathways within the inverse Collatz graph. A central objective is to distinguish genuine arithmetic effects from properties arising purely from modular geometry. The computational analysis therefore includes controlled comparisons across different initial distances and number classes, providing a framework for evaluating the observed patterns without assuming that primality is their cause. The paper brings together theoretical derivations, numerical experiments, and a reproducible computational implementation. Its results offer a structured perspective on the interaction between modular arithmetic, trajectory convergence, and the global organization of the Collatz graph. The complete mathematical derivations, experimental datasets, comparative analysis, and implementation are provided in the full paper. Package Contents Complete research paper (10-page PDF). LaTeX source and vector figures. Self-contained C++17 computational reference implementation. CC BY 4.0 license for the paper and MIT License for the software. Keywords Collatz conjecture; 3x+1 problem; Syracuse map; 2-adic arithmetic; modular coalescence; trajectory convergence; prime distribution; parity dynamics; discrete dynamical systems; spectral analysis; Markov chains; Mersenne numbers; inverse Collatz graph; computational number theory.