An Information-Theoretic Reframing of the Collatz Conjecture: From Heuristic Frequencies to Algorithmic Randomness

This paper presents a novel framework for analyzing the Collatz conjecture by reformulating the problem within the domain of symbolic dynamics and algorithmic information theory. We model the iterated application of the odd-step Collatz map, $T^{*}(n) = (3n+1)/2^{\nu_{2}(3n+1)}$, as a deterministic, confluent rewrite system. By applying the entropy-complexity correspondence for such systems, we establish a precise relationship between the algorithmic complexity of an integer and the information-theoretic properties of its trajectory. The core contribution of this work is the replacement of heuristic statistical assumptions, such as the Uniform Dip Frequency Conjecture, with a more fundamental principle: the \textbf{Algorithmic Randomness Hypothesis (ARH)} for Collatz valuation sequences. We posit that the sequence of 2-adic valuations generated by a typical orbit is algorithmically random and thus incompressible. Under this hypothesis, we formally prove that the expected logarithmic potential of an orbit exhibits a negative drift, compelling all trajectories to be bounded and, by extension, to terminate at 1. This reframing connects the notorious difficulty of the Collatz problem to the concept of computational irreducibility and quantifies the information loss, or "one-wayness," of the map via its symbolic degeneracy, $H(Y|X)$.