From Interpolation Amplitudes to Zero Geometry: Local Universality in a Critical Schur Interpolation Problem

This paper introduces a zero-geometric framework for a family of finite Schur interpolation problems. The central result is an exact factorization theorem showing that every observable studied in the manuscript factors through the zero set of the interpolating polynomial obtained after Schur reduction, identifying the finite zero process and its associated complex resolvent field as the natural mathematical objects of study. Building on this framework, the paper develops a locally rescaled zero process, an exact resolvent identity, and a deterministic harmonic tail decomposition. Extensive numerical investigations reveal dimension-stable local behavior of the imaginary part of the rescaled resolvent, while the real part exhibits a separation into local and macroscopic contributions. The manuscript also documents several natural normalization strategies that fail, thereby helping delimit the asymptotic structure of the problem. The work combines exact mathematical results with reproducible computational evidence and is intended as the first step toward an analytical theory of zero geometry, local universality, and resolvent fields arising from Schur interpolation problems.