SMALLEST PERIOD IN LOW COMPLEXITY SUBSHIFTS

The paper proves Theorem 5.1: for a low-complexity subshift X determined by a shape D and patterns P with |P| ≤ |D|, if every x in X is 1-periodic (a good pair), then the smallest period α(x) is uniformly bounded across X. The proof uses a universal annihilator, a bootstrapping lemma, and a compactness/limit argument to derive a contradiction if the bound were unbounded, and is consistent and complete in the manuscript (definition of good pair and Theorem 5.1, plus Lemma 3.2) . The candidate model solution attempts a different, combinatorial proof, but it hinges on an unproven and, as stated, generally false injectivity claim about overlap maps (using L_t and R_{-t}) to enable “free substitution” along an edge. That claim does not ensure local consistency along t-edges, so the subsequent propagation lemma is unsupported. Hence, the model’s solution has a fundamental gap, while the paper’s argument is sound.