ASYMPTOTICS OF TRANSVERSALITY IN PERIODIC CURVES OF QUADRATIC RATIONAL MAPS

The paper’s Theorem A states exactly χ(Sp) = 2/3·η′(p) − η′II(p) for p ≥ 3, identifies S1 ≅ C and S2 ≅ C*, and gives χ(Ŝp) = Np + 2/3·η′(p) − η′II(p) . Its proof uses a meromorphic one-form dτ on Ŝp and a Main Lemma (an asymptotics-of-transversality identity) to convert local orders at punctures into the Euler characteristic; type II centers are counted via zeros of G and puncture multiplicities are related to η′(p) by Bezout identities . The candidate solution reaches the same formulas via a different route: a holomorphic renormalization/index map Θp from Sp to the (2,3,∞) modular orbifold and an orbifold Gauss–Bonnet/Stokes count with defects attributed to polynomial parameters and type II centers. While the model’s approach is coherent and standard in spirit, it asserts (without full proof) global holomorphicity of Θp, that only those two defect types occur, and it implicitly assumes connectedness of Sp (which remains open in general per the paper’s introduction) . Net: same results, distinct methods; the paper is rigorous and complete, the model is essentially correct but omits some justifications.