Topological Entropy of Turing Complete Dynamics

The paper proves that any regular Turing machine has positive topological entropy by building two φ-cycles with the same shift direction, concatenating the induced micro-blocks u1 and u2, and obtaining an exponential lower bound on |S(n, R_T)| (Theorem 4). The candidate solution constructs the same objects (micro-blocks per pair, macro-blocks per cycle), codes them along the ε-ray of the tape, and derives the identical exponential bound, differing only in notation (T_max vs. a) and a slightly more explicit alignment argument. No substantive logical gaps appear in either; the arguments are essentially the same.