Encoding subshifts through sliding block codes

The uploaded paper’s Theorem 1.1 states exactly the equivalence in question and proves it under the hypotheses that X is a mixing SFT, Y is a mixing sofic shift, and π: X→Y is a factor code, with r_n(π) defined as the number of n-periodic points in Y that have an n-periodic preimage in X (Definition 2.1) . The paper also presents the elementary necessity (“only if”) argument via injectivity on orbits, matching the model’s (1⇒2) step . For sufficiency (2⇒1), the model appeals directly to the very theorem established in the paper (MacDonald 2022), whereas the paper supplies a full constructive proof using markers, blanks, and stamps (see the proof sketch and the key intermediate results: Proposition 3.5 and the quantitative inputs in Section 4, including Proposition 4.3) . Thus, both are correct; the model’s solution is non-constructive (by citation), while the paper’s is constructive.