BUFF FORMS AND INVARIANT CURVES OF NEAR-PARABOLIC MAPS

The paper’s Theorem B (Tameness) states exactly the two conclusions at issue—(i) landing of γ_n at the repelling fixed point selected by the sign of |λ_n|−1, and (ii) uniform (hence Hausdorff) convergence of γ_n to γ—under the non‑tangential hypothesis λ_n^q→1 (equivalently μ_n→1), and gives a complete proof using Buff forms and a rectifying coordinate that lifts the dynamics to near-translations, together with homotopy and trapping arguments (see Theorem B and its setup in the PDF, including the near‑parabolic normalization f=g^∘q and Λ_n, M_n parameters ; the rectifying coordinate and lifted dynamics F(Z)=Z+1+u_f are developed in §3–§4 ; the non‑tangential geometry that separates the two cases and pins down On is in §5.1 ). By contrast, the model’s solution attempts a classical near‑parabolic/Fatou‑coordinate proof but makes a critical leap: from Φ_n(γ_n(t+1))=Φ_n(γ_n(t))+1, it asserts that Φ_n(γ_n(t))=t+C for all t≤0. That inference is unjustified: H_n(t):=Φ_n(γ_n(t))−t is merely 1‑periodic, not necessarily constant. Without establishing that H_n is constant (or choosing a parametrization that forces it), the “constant Ecalle height” and landing conclusions do not follow as written. Additional nontrivial steps are also only sketched (continuous choice of petals P_n and coordinates Φ_n with kernel convergence; compact convergence of Φ_n^{-1} on left half‑planes). Thus the paper is correct and complete on these claims, while the model solution contains a key gap and missing technical justifications.