Regularity properties of the α-Wilton functions

The paper proves W_α ∈ BMO exactly on [1−g, g] (Theorem 1.1) and shows optimality by constructing rational parameters with failure of BMO to the right of g and arbitrarily close to 1−g from the left (Theorem 1.2) using matching and a type A/B singularity classification; the main positive step is the uniform bound W_α − W_{1/2} on [1−g, 1/2], achieved via a four-state comparison of the A_α- and A_{1/2}-orbits and continuant estimates (Proposition 4.4, Theorem 4.1) . The candidate solution states the correct final results but replaces the essential bound W_α − W_{1/2} with an unsubstantiated claim W_α − W_{1−α} ∈ L^∞; this is neither stated nor proved in the paper and the cited argument actually compares α with 1/2, not with 1−α . On the negative side, the candidate asserts a type A singularity at 0 for all α ∈ (g,1], which contradicts the paper’s explicit remark: for α < 1, 0 is the prototype of a type B singularity (the type A singularity occurs at the rational parameter α itself when the matching index is odd) . A minor mismatch also appears in the explicit sequence of pseudocenters u_m accumulating at 1−g (the paper uses [0; 2, 1^{2m−1}]) . In sum, the paper’s results and methods are correct; the model’s final conclusions align with the paper but rely on a key incorrect substitution and a misidentified singularity location.