On the Linear Complexity Associated with a Family of Multidimentional Continued Fraction Algorithms

The paper’s Theorem 1 proves exactly the bound 2n+1 ≤ p(n) ≤ 3n for the triangle map’s S-adic languages under the stated rational-independence hypothesis, via a careful bispecial-factor analysis and a Gauss-coding/antecedent framework. The candidate solution asserts the same bound but (i) replaces the lower bound proof with an unsubstantiated reduction to a 3-interval exchange coding, and (ii) gives an incorrect/insufficiently justified counting argument for the upper bound based only on a right-marking property. The paper’s argument is complete and correct; the model’s proof is not.