Structural Bounds and Entropic Decay in the Collatz Process: A Formal Framework

We present a rigorous analysis of the Collatz process, introducing structural and probabilistic arguments that eliminate the possibility of infinite divergence and constrain the formation of non-trivial cycles. By isolating the odd-only dynamics through the maximal odd iteration function T∗(n)T^*(n)T∗(n), and proving a generalized Power-of-p Attainment Theorem, we show that all trajectories must eventually collapse. We define a novel entropy metric for measuring directional bias in recursive modular functions (RMFs), and demonstrate that the Collatz process exhibits consistent downward pressure. Combining these techniques, we conclude that the only remaining route for non-convergence lies in the existence of statistically forbidden cycles.