Factor Complexity of the Most Significant Digits of a^{n^d}

The paper proves that for w defined by the most significant decimal digit of 2^{n^d}, the factor complexity p_w(k) is eventually a polynomial of degree d(d+1)/2, via a d-dimensional affine unipotent torus model and a careful hyperplane-arrangement count. The candidate solution claims essentially the same end-result but (i) asserts a torus of dimension D = d(d+1)/2 (confusing the ambient system dimension with the complexity polynomial’s degree), (ii) generalizes to arbitrary base b and multiplicatively independent a without justification, and (iii) glosses over key technical steps (density/equidistribution of the orbit for the proposed system, control of singular intersections) that the paper handles rigorously. Because of the dimension mistake and missing arguments, the model’s solution is not correct as stated, even though its conclusion matches the paper’s theorem in the special case a=2, base 10.