Circle homeomorphisms with breaks with no C2−ν conjugacy

The paper’s Theorem 2 proves that for f ∈ B^3(p,c) with irrational rotation number and negative Schwarzian, and for the linear fractional model g with the same rotation number, there is no C^{1+α} conjugacy for α > 1 − ν(c) (stated explicitly as not C^{2−ν(c)}), with ν(c) depending only on c. The proof uses a cross-ratio-based functional ξ_f, additive under composition, together with a negative Schwarzian bound ξ_f(J) ≤ −s|J|^2 (Lemma 2), Hölder control under conjugacy ξ_h(J) = O(|J|^α) (Lemma 1 and Corollary 2), and renormalization-driven exponential bounds on dynamical partitions (Theorem 3), culminating in Section 6’s derivation α ≤ 1 − ν(c) (see Theorem 2 and the argument around (22)). By contrast, the candidate solution relies on a two-interval cross-ratio Δ and asserts (i) an upper bound of order |I|^{1+α} for Δ_h and (ii) an equality Δ_f = Δ_h (since g is Möbius), both of which overlook the extra Δ_h(f(I),f(J)) term coming from composition and contradict the paper’s established estimate ξ_h(J) = O(|J|^α). It also misattributes Section 6’s inequality (22), which concerns sums of squares of lengths, to a direct lower bound on a single Δ_f(I,J). These gaps make the model’s proof outline incorrect, while the paper’s argument is complete and correct.