A Rigorous Framework for the Collatz Conjecture: p-Adic Dynamics and Spectral Analysis

This paper develops a comprehensive analytical framework for studying the Collatz conjecture through its natural extension to the 2-adic integers Z2. We establish rigorous foundations for the 2-adic Collatz dynamical system, proving that the extended function T is continuous and preserves the normalized Haar measure μ. Our main contributions include: (i) explicit computation of transition probabilities between cylinder sets, (ii) detailed spectral analysis of the Koopman operator on invariant subspaces, and (iii) characterization of the precise obstacles to proving unique ergodicity. While we do not claim a proof of the Collatz conjecture, we provide substantial evidence for the ergodic hypothesis and develop novel techniques that offer promising avenues for future research. Numerical simulations support our theoretical findings and illustrate the intricate structure of the Collatz dynamics in the p-adic setting.