An infinite-dimensional Kolmogorov theorem and the construction of almost periodic breathers

The paper’s Theorem 2.1 establishes a Kolmogorov-type result on the thickened infinite torus T∞_σ under Bourgain’s Diophantine nonresonance, using a generating-function-based KAM scheme with a carefully designed contraction sequence and super-exponential convergence. It proves the existence of an exact symplectic triangular map x = u(ξ), y = v(ξ) + ⟨u_ξ(ξ)^{-1}, κ⟩ mapping D_{σ/4,σ/4} into D_{σ,σ}, with W_ξ(ξ,0) = 0 and W_κ(ξ,0) = ω, and obtains quantitative bounds of order O(1/K) (specifically, ∥ϕ−id∥ ≤ 2 σ^{2/η}/K and ∥W_{κκ}−Q∥ ≤ 2 σ^{2/η−1}/K). The candidate solution instead proposes a Newton–KAM parameterization scheme with fixed analyticity loss, claims convergence after only three steps while also asserting exact satisfaction of the Kolmogorov conditions, and derives bounds of order O(e^{-K})—all of which conflict with the paper’s precise iteration design and final quantitative estimates. Furthermore, the candidate’s stepwise triangular resolution (e.g., setting h = 0 when solving for the constant part k̄) is not justified, and the domain-loss accounting is inconsistent with the need for infinitely many KAM steps to reach exact conjugacy.