The Collatz Conjecture is a Two Bit Problem

In this work we prove a reduction of the Collatz Conjecture to an open question of ergodic mixing. By positioning the Collatz sequence as a race between the leading bit and trailing bit of the iterates in dyadic representation, we prove that convergence hinges on the local ergodicity of just two "indicator" bits governing contraction. We prove a structural upper bound on the velocity of the leading bit, and a stochastic lower bound on the velocity of the trailing bit contingent on the uniformity of these indicator bits. The resulting velocity gap implies global convergence for any trajectory meeting the local ergodicity condition. We derive the asymptotic uniformity of these bits via the fractional position map, and assess the sensitivity to distributional distortion. We conclude that the Collatz Conjecture is false if and only if the invariant measure of the underlying dynamical system collapses to a singular support, thereby reducing the number-theoretic problem to a single, testable proposition in ergodic theory: prove that the indicator bits are strongly mixing.