Deterministic Limits and Ergodic Properties of a Bitwise Syracuse Map

        A recent work \cite{Galoppo2024} proposed a novel framework for studying the Syracuse function and finite-state machine capturing its dynamics. In this paper we extend that work, giving deterministic proofs that non-convergent simple cycles have finite limits. We further investigate the non-uniform distribution of fractional positions of iterates observed in \cite{Galoppo2024}, prove that these asypmtotically follow Benford's Law, and that the low-order bits are uniformly distributed in the limit. This result formally proves that the Collatz conjecture is equivalent to a single, unproven ergodic property: the strong mixing of the low-order bit sequence. We provide strong empirical evidence for this mixing property and conclude by framing the full conjecture as a specific, open problem in the study of perturbed dynamical systems.