Layered Grouping of Numbers According to Step Depth in Collatz Dynamics

Abstract

In this study, a systematic classification of odd integers in Collatz trajectories according to their step depths and modular layers is presented. Odd numbers are categorized into n-th Groups and ROOT Groups based on the step count required to reach a known node value after the initial 3x+1 transformation. This approach demonstrates that the entire set of odd numbers can be covered by 2^n modular steps and that trajectories can be reduced to recurring sub-dynamics.