A Rigorous Framework for the Collatz Conjecture

We analyze Collatz by restricting to $A = \{N \text{ odd}: v_2(3N+1) \le 6\}$, containing $63/64$ of odd integers. We prove equivalence to full conjecture, characterize $A$ via congruences, establish growth constraints, develop Markov models, and synthesize with Tao's result and 2-adic theory.