On Structure of Topological Entropy for Tree-Shift of Finite Type

The paper’s Theorem 5.1 constructs a layered strongly connected graph G_{y_l;N} with |V_i| = n_i and edges from V_i to V_{i+1}, deriving the exact product p_{G;1}(n) and the limit h(T_G) → μ(d)_{y_l}, then proving density in [(d−1)log 2, ∞) via a careful control of cyclic shifts (notably y_1 ≥ 2^d and restricted digits) . The candidate solution replaces this by a “base–clone” two-layer gadget. That gadget forces every other tree layer to be inert (outdegree 1), so its layer-by-layer counting formula log N(n) = ∑ d^{t+1} log w_{i+t} is incorrect for the proposed graph; the ensuing error bounds and the limit μ(d)_{y_l} therefore do not follow. In addition, the model’s density step does not justify that the maximizing cyclic shift equals the chosen digit order, a point handled explicitly in the paper.