Word complexity of weakly mixing rank-one subshifts

The paper proves the four claims (weakly mixing examples with lim sup p(q)/q < 1.5+ε and with p(q) < q+f(q) infinitely often; and for totally ergodic rank-one, lim sup[p(q)−1.5q]=∞ and lim inf[p(q)−q]=∞; plus optimality via totally ergodic constructions with p(q) ≤ 1.5q+f(q) eventually) as stated in its Introduction (Theorems A–D) and carried out in Sections 4–5 using explicit rank‑one constructions and detailed right‑special word counts (e.g., equations (†)–(‡) in Section 4) . By contrast, the model’s solution relies on key incorrect or unsupported steps: it asserts that total ergodicity forces infinitely many distinct spacer lengths, which is contradicted by the paper’s Proposition 2.13 showing one can represent such systems (when lim sup p(q)/q < 2) with only two spacer values 0 and d occurring infinitely often; it also invokes an unsubstantiated equivalence “weak mixing ⇔ total ergodicity” for canonically bounded rank‑one systems, whereas the paper supplies its own weak‑mixing proofs without that equivalence. Additionally, the model’s complexity bounds are argued via a heuristic density-of-s(q)=2 argument with no rigorous control of the exceptional ranges, while the paper provides precise counts at anchor lengths. Hence, while the model’s end statements mirror the paper’s theorems, its proof contains substantive errors and gaps, so the correct verdict is that the paper is correct and the model proof is not.