KAM THEOREMS FOR MULTI-SCALE TORUS.

The paper proves a multi-scale KAM theorem for Hd(x,y,θ,η,ϕ,I) = Nd(y,η,I) + ε P(x/λ1, y, θ, η, λ2 ϕ, I) with λ1 = ε^α, λ2 = ε^β, giving: (1) persistence of invariant 3d-tori under (R), (2) partial frequency preservation under (K), and (3) isoenergetic tori preserving n frequency ratios under (R)+(K)+(Iso) with the stated β-bound; see Theorem 1.1 and Section 5 for the multiscale setup and iteration scheme . The model’s solution uses a canonical rescaling to convert weighted small divisors into a standard Diophantine condition on ρ(b) = (ω/λ1, Λ, λ2 Ω) and then runs a standard analytic a posteriori KAM/Newton scheme, also handling partial and isoenergetic constraints. This is mathematically equivalent in spirit to the paper’s weighted small-divisor formulation and homological equations (e.g., (5.8)), just presented in a different gauge with the same β restriction and the same three conclusions . Minor differences are present (e.g., the model states an explicit O(ε^σ) measure loss and “Whitney-analytic” dependence, while the paper asserts the relative measure tends to 0 and proves Whitney C^{d−1} regularity), but these do not constitute substantive conflicts with the main results .