Optimal linearization of vector fields on the torus in non-analytic Gevrey classes

The paper’s Theorem A states quantitative Gevrey linearization with the precise constants c(a,κ)=32^b·(κ−1)(κ^{1−a}−1)^{−1/(1−a)} and exponent ι=8^{−b}(2r)^{−1}υ (υ=8^{−b}(r−cτ)), under ω∈A^a_{τ0} and smallness ε≤γ_{τ0}(ω)ε*; see the statement of Theorem A and its parameters (3.5), and the final estimates (4.26)–(4.27) in the proof . The proof strategy—Gevrey→analytic approximation via Popov and an analytic KAM step à la Rüssmann–Pöschel with explicitly optimized Ψκ(σ) giving Ψκ(σ)≤e^{δτϕ_b(τ/σ)} and yielding c=32^bδ—is developed in §4.2–§4.4 (see (4.10), (4.12), and (4.18)) . The candidate solution follows this scheme closely (same classes Fa_r and A^a_τ, same constants b, δ, c, budget r>cτ, accumulation of conjugacies, and the final ι), so both are correct and essentially the same proof architecture.