Some remarks on the Classical KAM Theorem, following Pöschel

The paper’s Theorem A states and proves (with a corrected KAM step) a parameter-dependent Herman normal form for H = N + P with N(I, ω) = e(ω) + ω·I, under |P| ≤ γ α r s^ν and α s^ν ≤ h, yielding a Lipschitz parameter map ϕ and embeddings Φ with the weighted bounds (1.2) . Its proof carefully controls the averaged quadratic-in-I terms via a refined polynomial approximation and a linear-convergence scheme (Proposition 2.1 and the iteration) . The model’s solution aims at the same result but, in the one-step estimate for P^+, it omits the contribution of the averaged quadratic term ⟨T_K P⟩^{≥2}. Without bounding this O(|I|^2) term on the shrunk domain, the claimed “quadratic” reduction |P^+| ≤ C e^{−Kσ}|P| + C(α r σ^ν)^{-1}|P|^2 is not justified; the paper explicitly addresses this gap by replacing crude truncation with a polynomial approximation affine in I and by using a linear scheme that yields |P^+| ≤ κ ε (κ ≪ 1) per step (2.1)–(2.3) . Hence the paper’s argument is correct, while the model’s proof is incomplete.