KAM below Cn

The paper proves a “Differentiable KAM Theorem” for rotational flows on T^n: if ω is Diophantine with exponent τ and the perturbation P is small in the block-L2 Besov-type norm ‖P‖_{τ+1,b}, then there exist a constant vector field Y and a C^1 diffeomorphism Ψ with Ψ^*(N+P−Y)=N; see the statement and the norm definition early in the paper and the iterative scheme in Sections 3–5 . The candidate solution reaches the same conclusion under the same smallness condition, but via a different route: a Littlewood–Paley/Nash–Moser KAM-by-blocks scheme directly in B^{τ+1}_{2,1}, solving the cohomological equation at each step and controlling commutators and remainders. The paper’s proof uses analytic smoothing, a Step Lemma based on a finite-dimensional fixed-point argument, and an iteration with analytic-width bookkeeping; the model uses Besov algebra/commutator estimates and flow pullbacks. Both approaches are logically coherent and compatible with the Diophantine loss τ and the need for one extra derivative to control the flow. Minor details the model leaves implicit (e.g., explicit constants, bounds ensuring the overlap between low-pass tail and sup-norm constraints, and rigorous justification of the Lie-series in the B^{r}_{2,1} setting) are standard and fillable using known Besov/Kato–Ponce machinery. Hence, both are correct, with different proofs.