A note on the perturbations of subshifts

The paper proves the 1/n upper bound for the entropy drop in irreducible sofic shifts (Theorem 1.2) by passing to a right‑resolving sofic cover and reducing to a multi‑word perturbation of an irreducible SFT, where a known complexity-based bound gives h(X)−h(Xw) ≤ C/n; via a counting argument the perturbed SFT and sofic entropies are shown equal, yielding the result. This is stated and proved succinctly in the paper (Theorem 1.2 and its proof via Theorem 3.2) . By contrast, the model’s proof incorrectly treats the (n−1)-block presentation of a general sofic shift as a conjugate 1‑step SFT, identifies h(X)=ln ρ(A_n) and h(Xw)=ln ρ(A_n−E_w), and equates a Parry cylinder weight with ℓ_p r_q/ρ(A_n). Those steps are only valid for SFTs, not for general sofic shifts; in general the higher‑block edge shift is a strict superset of X, and deleting a single edge there does not compute h(Xw). The model also ignores multiple preimages of w under a right‑resolving cover (the paper handles this explicitly), so the spectral perturbation it performs does not control the entropy of Xw. Hence the paper’s argument is correct, while the model’s is flawed.