Improved Gevrey-1 estimates of formal series expansions of center manifolds

The paper proves the limit (−1)^n ϕ_n / Γ(n+a) → S∞ rigorously (Theorem 1.1), by reducing to a ≥ 2 via a normal-form/blow-up step, formulating the Borel-plane equation (w+1)Φ + a⋆Φ = F0 + Σ F2,l⋆Φ⋆l, inverting the linear operator (w+1)+a⋆ (Lemma 4.1), constructing a Banach space that controls the singularity at w = −1, establishing an initial Γ(n+a)-type bound (Lemma 4.10), and bootstrapping to the full limit (Section 5) . By contrast, the candidate solution contains a key algebraic slip when differentiating in the Borel plane (it incorrectly defines H = ∂ξF̂ − a φ̂ instead of H = ∂ξF̂), and omits the normal-form step used to avoid logarithmic issues when a ∈ −ℕ, as well as the quantitative estimates needed to ensure the subdominant contributions are o(Γ(n+a)) (the paper’s Remark 3 explains the log phenomenon and why a ≥ 2 is enforced) . The model’s approach is directionally aligned but not a complete or correct proof as written.