Noncommutative Coboundary Equations over Integrable Systems

The paper proves that for the integrable base f(θ,I)=(θ+I,I) and an analytic family η_ε with η_0=Id, the following are equivalent: (A) periodic-orbit condition (POC), (B) existence of a formal power-series conjugacy, and (C) existence of an analytic conjugacy φ_ε, see Theorem 3 and Definition 2 with estimate (1.7) , and the proof overview explaining (C)⇒(A),(B), (B)⇒(A), and especially (A)⇒(C) via a Nash–Moser scheme . The linear commutative step (Lemma 19) constructs a solution to α∘f−α=β under the Fourier-coefficient obstruction and provides tame estimates with a loss of domain . The nonlinear step (Proposition 24) implements an iterative correction with explicit bounds, again with controlled loss of analyticity width, yielding convergence to an analytic conjugacy . By contrast, the model’s core step (A)⇒(C) invokes the standard analytic implicit function theorem on fixed Banach spaces, asserting that the linearized operator L: X_s→Y_s is a bounded isomorphism “via Fourier/Weierstrass,” and hence a direct IFT applies. This misses the crucial loss-of-domain phenomenon: Lemma 19 only yields an inverse with shrinkage of the complex width (‖α‖_{ρ−δ} ≤ c δ^{−(d+1)}‖β‖_ρ), so L does not have a bounded inverse X_s→Y_s on the same domain. The paper therefore uses a Nash–Moser/KAM-type iteration with truncations and rescaling in ε to overcome this loss (see the iterative scheme (3.1)–(3.7)) . The model’s alternative linear step via Weierstrass division is fine in spirit (it matches the paper’s removable-singularity/one-variable argument near resonances), and its (C)⇒(A),(B) and (B)⇒(A) arguments align with the paper’s remarks . But the decisive nonlinear step is invalid as stated, because the required inverse for L on a fixed analytic domain does not exist; a hard (Nash–Moser) implicit function theorem is needed, exactly as implemented in the paper.