Parabolic Components in Cubic Polynomial Slice Per1(1)

Zhang proves Theorem A—every parabolic component boundary in Per1(1) is a Jordan curve—via parameter puzzles and a three-case local-connectivity analysis (Misiurewicz parabolic, non-renormalizable, renormalizable), culminating in a Carathéodory extension and an injectivity argument (Proof of Theorem A) after setting up the parameterization and puzzle machinery (including Proposition 2.3.6 on the Φ-preimage W and its boundary) and constructing internal/parameter rays with landing properties (e.g., Proposition 3.2.11 and rational rays landing) . The shrinking/complex-bounds input is encoded by non-degenerate annuli whose moduli sum to infinity unless renormalization occurs (Theorem 3.4.5) , and Jordan boundaries of immediate basins used in the conjugacy arguments are attributed to Tan–Yin as cited by Zhang . The candidate’s solution follows the same para‑puzzle blueprint—holomorphic motion of puzzle graphs, nested parapuzzle pieces with divergent moduli, Carathéodory–prime ends—differing mainly in its Ecalle‑height description of boundary arcs, which Zhang does not emphasize but which is compatible with his parametrization framework. Hence both are correct, with substantially the same proof architecture.