Quasi–periodic motions in generic nearly–integrable mechanical systems

The paper’s note states Theorem 1 exactly as the model proves it—namely, for H(y,x;ε)=|y|^2/2+ε f(x) with f in the generic class P^n_s, the complement of maximal KAM tori has measure ≤ c ε |log ε|^{11n+4}—and outlines the same mechanism: resonance decomposition, pendulum-type normal forms near simple resonances, control of double resonances, and a quantitative KAM step relying on Kolmogorov non-degeneracy in action–angle variables for the secular Hamiltonian H_k. The model’s solution reproduces this standard pipeline, including the role of the P^n_s hypotheses (Morse with distinct critical values and a lower bound on high modes) and the optimization K~const·|log ε|/s, yielding the stated measure bound. Differences are present only in level of detail (the paper is a short note pointing to a 101-page preprint) and in bookkeeping constants; conceptually and technically, they are aligned.