STRONG CLOSING LEMMA AND KAM NORMAL FORM

The paper explicitly states and sketches a proof of the claimed uniform-in-ε periodic-orbit result for H(I,θ)=ω·I+εf with f∈C^r_c(T* T^n) and r large, via Dirichlet rational approximation, a KAM-type normal form, and a Lyapunov-center continuation, see Theorem 1 and the proof outline (normal form, energetic reduction, and Lyapunov step) in the text. However, key ingredients are only tersely justified: the “generic” Diophantine condition (GC2) on the small frequency vector ̟ is asserted as ‘full measure’ but not shown to hold Baire-generically in f, let alone uniformly in ε, and the normal-form/KAM iteration estimates are presented without full quantitative control or a uniform-in-ε Diophantine constant. There is also a minor but real gap in explaining why the resulting orbit necessarily intersects supp f (the proof refers to a maximum of f where the argument has averaged to F). Thus the paper’s argument is promising but incomplete. By contrast, the model’s claim that the problem was likely open overlooks this 2022 preprint asserting precisely the desired statement, as well as its concrete mechanism, so the model answer is also incomplete. The appropriate verdict is that both are incomplete, albeit for different reasons. See the paper’s statement of Theorem 1 and its proof steps for details.