PRIMITIVE TUNING VIA QUASICONFORMAL SURGERY

The paper’s Main Theorem explicitly states that for a primitive postcritically finite hyperbolic polynomial f0 with a fixed internal angle system, the IK straightening map χ: C(f0) → C(T) is bijective and C(f0) is connected (see the Introduction/Main Theorem and §2.4 for the set-up and definitions) . The surjectivity is proved via a carefully constructed quasi-regular surgery supported on a Yoccoz puzzle, together with a convergence theorem for Thurston’s algorithm and Kahn’s quasiconformal distortion principle (§§3–6) . By contrast, the model’s proposed inverse T(g) is defined by replacing the time-1 dynamics on each U_v with the r(v)-step polynomial-like return map g_v in a single iterate, and then “rectifying” by an invariant Beltrami equation. This breaks the time-1 dynamics and does not ensure continuity across ∂U_v: the boundary conjugacy they enforce uses f0^{r(v)} on ∂U_v, while the outside branch of their global map is f0 (one iterate), so the two sides need not match on ∂U_v. Moreover, the existence and uniqueness of a global invariant Beltrami solution with the claimed properties is not justified. The paper avoids precisely these pitfalls by proving a conjugacy outside deep puzzle pieces and then running Thurston’s algorithm to obtain a polynomial f with the required λ(f0)-renormalizations (Proposition 3.2 and §6) . Hence, the paper’s result and proof are correct and complete for the stated setting, while the model’s construction is flawed.