NON-STATIONARY NORMAL FORMS FOR CONTRACTING EXTENSIONS

The paper’s Theorem 4.3 is stated and proved under Assumptions 4.1 with the precise smallness parameters ε0(χ) from (3.13) and the additional Hölder-regularity constraint (4.2): ν = χ1 − (N+α)χℓ > 0 and ε < ν/(N+α+1) . Its proof builds jets using a contraction on the quotient bundle Q(n)/S(n) (equations (5.4)-(5.7)) and the Green-series representation compatible with Lemma 5.1’s bounds, which are controlled by λ and ε0(χ) . For the remainder, the paper does not invert the linear cohomological operator A = Lh − h∘L; instead, it constructs a nonlinear graph transform T and proves it is a contraction in a C^{N,α} Banach space using δ = ν − (N+1+α)(ε+ε′) > 0 (see (5.19)–(5.24)) . By contrast, the model’s Step 2 incorrectly claims A is invertible by a forward Green series based on the “uniform negativity” of (χ1 + ε) − (N+α)(χℓ − ε); under (4.2) this quantity equals ν + (N+α+1)ε, which is positive, not negative, so the proposed series does not converge. The remainder of the model aligns conceptually with the paper: sub-resonant jets up to degree d = ⌊χ1/χℓ⌋, elimination of strict sub-resonances using μ via a contraction for Φ^{-1} (Lemma 5.5), uniqueness up to Sx or Rx, and the centralizer description (parts (1′),(2),(2′),(3)) . Hence the paper is correct, but the model’s core Step 2 is flawed.