Perturbing Subshifts of Finite Type

The uploaded paper proves an entropy lower bound for SFTs perturbed by forbidding a finite set S of words. Precisely, Theorem 1 asserts that if every word in S has length at least k and there exists t in (1, λ^k) with r = (1 + k C λ^{2k} p(t^{1/k}/λ))/t < 1, then h(Σ) − h(Σ⟨S⟩) ≤ −(1/k) log(1 − r). The proof proceeds via Parry measure, a weight function w(σ), a good-extensions argument (Lemma 3), and a blocking argument to bound pk and permit t up to λ^k, yielding the explicit λ^{2k} factor (Theorem 1 and its proof via Proposition 1 and Equation (5)) . The candidate solution derives the same inequality and the same r using a different route: a blockwise union/Markov inequality under the Parry (maximal-entropy) measure, plus a uniform conditional cylinder bound and an iteration across blocks, then converting measure to counts via uniform cylinder bounds. The constants and the λ^{2k} factor appear in a way consistent with the paper’s final r. Minor issues in the model’s writeup (e.g., attributing a λ^{2k} multiplier to “unknown right context” despite the Markov property) do not invalidate the conclusion and simply introduce slack. Overall, both the paper and the model reach the same bound; the paper’s method is more structured (via w(σ) and pk), while the model’s is a streamlined probabilistic alternative.