ENTROPY BOUNDS FOR GLASS NETWORKS

The paper proves lim_k h(J[2](TGr(k))) = h(φ(D_TR)) by pushing J[2](TGr(k)) through a finite-to-one sliding block code ζ_k to a nested sequence of shift spaces X_Fk whose intersection is φ(D_TR), then invoking Lind–Marcus’s proposition on entropy continuity for decreasing chains; it also notes h(J[2](TGr(k))) equals log of the Perron eigenvalue of TGr(k)’s adjacency matrix, giving computable, decreasing upper bounds (Theorem 4.19, Proposition 4.17) . The model gives a valid alternative based on k-block language agreement L_n(Y_k)=L_n(X) for n≤k, submultiplicativity, and X⊆Y_k, which also yields convergence and the same computable bounds. Minor notational gaps (edge- vs node-coding; explicit use of the 2-block map; “last” vs “next” cycle memory) can be corrected without affecting correctness. Overall, both arguments establish the same result and are consistent with the paper’s constructions and entropy bounds .