On the linearization of analytic diffeomorphisms of the torus

The paper’s KAM proof hinges on anisotropic analytic domains and norms tailored to the weak-Bryuno construction, with a precise small-divisor estimate encoded by Φ(ℓ,β,δ) and the recursive radii δ_{β,n} and smoothing in β, culminating in Proposition 2 and Theorem 1.9; these are internally consistent and carefully supported by lemmas (e.g., Lemmas 3.10, 3.15–3.18) and the KAM step. The candidate’s argument, while close in spirit, incorrectly attempts to pass from the paper’s anisotropic linear estimate to an isotropic ℓ1-analytic norm bound for the cohomological solution u_n; the key inequality relating isotropic and anisotropic weights is reversed, breaking the claimed estimate ||u_n||_{δ_{n+1}} ≤ Γ_n ||w_n||_{δ_n}. It also omits the paper’s essential control of the inverse conjugacy term g_{n+1} and the slice-mapping mechanism needed under the weak-Bryuno condition.