ON THE FORMULAS OF MEROMORPHIC FUNCTIONS WITH PERIODIC HERMAN RINGS

The paper proves Theorem B rigorously by a tailored quasiconformal surgery that produces a cubic map with a p-cycle of Herman rings, arranges a super‑attracting p‑cycle containing 0, and then derives the explicit normal form fa,b(z)=u z^2 (z−a)/(1−a z)+b. Crucially, the paper analyzes the critical set and a separation property to force this specific Möbius factor, and shows how to make a arbitrarily large via a precise conjugacy L_λ with κ_λ>1 as λ→0. The candidate solution gets the high‑level existence ideas right (Yoccoz + Shishikura surgery), and the p=2 algebraic check is correct, but it makes a fatal leap in “Step 5,” asserting without justification that every cubic map with a double critical at 0 has the special form u(z−a)/(1−a z). That is false in general: one only gets b+z^2·(Möbius), and the paper’s extra work is needed to enforce the special constraint B=−AC for the Möbius matrix. The parameter-scaling asserted in Step 6 is also not derived correctly; the paper’s L_λ argument is the right way to make a large.