Collet-Eckmann Type Conditions and Conformal Welding of Unicritical Quadratic Laminations

The paper defines the combinatorial Collet–Eckmann (CCE) condition for quadratic laminations and proves Theorem 1.1: CCE yields, at geometrically many scales, (N,C,2^{-n}r)-nice gluing circuits (via Lemma 3.5), which by a known criterion (Theorem 3.2) produce a Hölder continuous conformal welding; removability of Hölder trees then identifies the welded dendrite as the Julia set of a unicritical quadratic polynomial p_c(z)=z^2+c (explicitly stated in Theorem 1.1) . The model’s solution follows the same pipeline—build quantitative barriers from nice gluing circuits; propagate them under z↦z^2; obtain Hölder welding; and realize the external class as a polynomial—though it phrases the modulus propagation using Kahn–Lyubich’s quasi-additivity/covering lemma rather than citing the welding criterion directly. Aside from an extra (unneeded) assertion that CCE implies a TCE-type expansion, the model’s reasoning aligns closely with the paper and achieves the same conclusion, so both are correct with substantially the same proof structure.