On qc compatibility of satellite copies of the Mandelbrot set: II

The uploaded paper proves that for any two satellite copies with the same denominator q, the induced Douady–Hubbard homeomorphism ξ_{p/q,p′/q} is quasiconformal (locally the restriction of K-quasiconformal maps with K depending only on q), see Theorem 1.1 and its stated reduction to Theorems 2.7 and 3.8 . The proof constructs uniform pre-exterior quasiconformal equivalences, develops a holomorphic-motion/Beltrami-disk framework in the Lyubich space of quadratic-like germs, and then transfers dynamical quasiconformality to parameter quasiconformality (Theorem 3.8) while using holomorphy on hyperbolic components to conclude global quasiconformality on each copy . By contrast, the candidate solution hinges on an unproved uniform a priori modulus bound m(q) for the canonical satellite renormalizations across an entire q-limb and on the claim that the straightening map χ_{p/q} itself is locally quasiconformal in parameter; neither point is established in the paper and the latter is not asserted there. Moreover, the candidate cites the very 2025 result under review to handle the parabolic root, which is circular. The paper’s method avoids these gaps via explicit pre-exterior constructions and holomorphic motions, whereas the model conflates dynamical QC straightening with QC regularity in parameter and assumes uniform complex bounds without proof. Hence the paper’s argument stands, while the model’s proof is incomplete/incorrect.