Moment-optimal finitary isomorphism for i.i.d. processes of equal entropy

The uploaded paper proves that any two finite-alphabet aperiodic (period-one) Markov shifts of equal entropy are finitarily isomorphic via a stationary code whose two-sided coding-radius tails satisfy P[R>n] = O(n^{-1/2} log^{O(1)} n), hence E[R^θ] < ∞ for all θ<1/2 (Theorem 0.1 and Theorem 3.13). The candidate solution asserts exactly this result and cites the same work; however, its proof sketch uses a different mechanism (blockwise bit extraction/injection and a nearest-neighbor bit-transport controlled by zero-drift random-walk hitting times), whereas the paper’s proof proceeds via marker processes, good systems of fillers, concentration inequalities, and “societies” matchings to achieve the n^{-1/2} tail. Thus the claims agree, but the proof methods differ substantially.