COMPUTATION OF DOMAINS OF ANALYTICITY OF LOWER DIMENSIONAL TORI IN A WEAKLY DISSIPATIVE FROESCHLÉ MAP

The paper’s setup and claims are consistent and modest: it formulates the invariance equation for the Froeschlé map (2.9) with the standard normalization (2.11), derives the order-by-order cohomology equations for the conservative (3.9) and dissipative cases (3.18)–(3.20), and computes the coefficients Rn using the automatic-differentiation recursion (3.5)–(3.8). All of these steps are standard and accurately stated, and the paper confines itself to numerical exploration of analyticity domains and Gevrey exponents (with figures illustrating the dissipative obstruction along |λ(ε)|=1 where λ(ε)=1−ε3). These points match well-known theory and are presented correctly and carefully . By contrast, the candidate solution overreaches on two key fronts. First, it asserts that the mere Gevrey-σ character implies 1/σ-summability and sectorial analyticity with optimal opening π/σ; that is not a consequence of Gevrey bounds alone and requires additional Borel-analyticity assumptions absent here. The paper does not make such a claim; it only provides numerical evidence about singularity patterns, e.g., the dissipative plots highlighting |λ(ε)|=1 curves, without asserting summability or optimal sectors . Second, the model claims that one can prove bounds of the form ∥g_n∥ ≤ A R^n n^{σ n} for any σ>2ℓ via a majorant argument. This is not supported by the paper’s theory (which cites [BlL23] for non-sharp Gevrey estimates) and appears inconsistent with the reported numerics showing larger effective slopes for some Diophantine frequencies (e.g., σ≈6.18 when ℓ=2), which would eventually contradict a uniform bound with a smaller σ . The rest of the candidate’s derivation (order-by-order solvability, inversion of Lω=Δ^2_ω using the Diophantine bound, linear growth of Fourier support, and the obstruction at |λ(ε)|=1 from small divisors 1−λ(ε)e^{-imω}) matches the paper’s setup and standard KAM folklore. But its claims about sectorial summability and the universality “any σ>2ℓ” exceed what is justified by the paper or standard references and are therefore judged incorrect or at least missing crucial hypotheses.