Finite-Scale Folding and Maximal Asymptotic Variance

Abstract

Based on the recent breakthroughs by OpenAI and consequently by Binder-Ivrii-Prause-Smirnov, we develop a finite-scale folding method for the nonlinearity of a schlicht function, based on a signed critical-point count for superharmonic potentials and a quantitative target cutoff. Furthermore, we use a cylindrical root transform to reduce the logarithmic cost of access to the point at infinity. Combined with a Littlewood-Paley formula as well as dilation, the method yields the maximal asymptotic variance bound $Σ_{\max}^2\le1$ for the unrestricted schlicht class of normalized univalent functions.