Sequence Entropy and Independence in Free and Minimal Actions

The paper proves both parts via Toeplitz constructions and independence theory, with rigorous upper bounds coming from odometer fiber estimates and Proposition 2.5 (sequence entropy equals log of the largest non-diagonal IN-tuple size). The candidate solution’s lower-bound constructions are plausible, but its upper bounds are flawed: (i) it asserts an “obvious” general bound h^* ≤ log m for m-symbol subshifts without justification; and (ii) in Part (2) it uses a factor map to claim an upper bound on sequence entropy, reversing the correct monotonicity (factors cannot increase sequence entropy). It also uses this to conclude there is no IN_{n+1}, which is circular without a correct upper bound. The paper’s arguments avoid these issues and are complete.