2-adic Structure and the Rigid Regime in Collatz Dynamics

This article develops a structural framework for the study of the Collatz dynamics based on its 2-adic structure. The approach introduces cylindrical constraints on the initial parameter, a binary budget describing the irreversible consumption of bits along dynamical excursions, and an autonomous return map governing the rigid regime. It is shown that any infinite orbit must eventually enter this regime, reducing the Collatz problem to a regularity condition on the parametrization of the entry map. The paper consolidates and reorganizes several results previously developed by the author in a series of related works into a single self-contained exposition.

Version 1.1 introduces minor corrections and improvements in the exposition. In particular, several statements concerning the rigid regime and the dynamical dichotomy have been clarified, and small formal adjustments have been made throughout the text.

Version 1.2 This version incorporates the results of Paper 9 in the series (2-adic Crystallization of the Ariadne Thread), published after version 1.2. Paper 9 closes Case A of the orbital gap unconditionally and reduces Case B to an explicit arithmetic criterion on the 2-adic expansion of the crystallized point. The abstract, introduction, conclusion, and results table have been updated accordingly.