Limits of Rauzy graphs of languages with subexponential complexity

The paper’s main equivalence theorem (Theorem 1.1 and again Theorem 2.8) states that the empirical spectral measures converge to the measure with density (1/π)√(4−x^2) on [−2,2]. This is not a probability density (it integrates to 2) and is not the spectral law of the infinite line Z. The correct limit law is the arcsine law with density 1/(π√(4−x^2)) on [−2,2]. See the itemized equivalences in Theorem 1.1 where the erroneous density is explicitly stated, and the same density is repeated in Theorem 2.8’s item (6) (both places: the statement as written is wrong) . Apart from this spectral misidentification, the rest of the argumentation (e.g., the use of the second moment to relate spectral convergence with p(n+1)/p(n), Lemma 2.6’s 1-in-1-out characterization, and the subsequence convergence Lemma 2.7) aligns with the model’s reasoning and is sound; for instance, equation (7) correctly relates the second spectral moment to 2·p(n+1)/p(n) up to O(1)/p(n) . The candidate solution fixes the spectral measure, proves convergence via closed-walk moments, and shows the full equivalence with the corrected arcsine law. Hence the correct outcome is that the paper’s statements are right modulo the explicit correction of the limit law; as stated, the paper is wrong on this point, whereas the model’s solution is correct and complete.