The multi-scale KAM persistence without a scaling order for Hamiltonian systems

The paper states a three-part KAM persistence theorem for multi‑scale Hamiltonians (Theorem 1.1) with a measure bound |bad set| = O(ϵ^{1/N}), under hypotheses (R), (K), (I), and an isoenergetic variant with a scaling factor t → 1 as ε → 0 . However, the nondegeneracy condition (R) introduces a symbol N that is never defined or used in the rank condition itself, yet it reappears in the measure bound O(ϵ^{1/N}), creating a mismatch between assumptions and claimed exponent . The proof outline via a parameterized KAM step (Proposition 2.1) is provided, but key quantitative linkages between the Diophantine threshold γ, the iteration parameters (r, K, σ, h), and the final ϵ^{1/N} measure exponent are only sketched; the Measure Estimate is described narratively without a worked derivation of the exponent N . There is also an explicit internal inconsistency: Theorem 1.1 asserts t → 1, whereas Remark 3 states t → 0 as ε → 0 , and the manuscript still contains placeholders (e.g., “Section ??”), indicating an incomplete draft. By contrast, the candidate solution gives a coherent a‑posteriori route: rescale by ε̃, invoke a Rüssmann–Łojasiewicz sublevel measure bound for resonances, choose γ ∼ √κ to match a standard analytic KAM smallness κ ≲ γ^2, and then obtain |bad set| = O(κ^{1/N}) with the correct frequency‑preservation statements (full, partial, and isoenergetic up to a factor t → 1). While concise, this argument aligns with established a‑posteriori KAM frameworks and remedies the paper’s missing definition of N by tying the exponent to the sublevel estimate.