Large-Window Asymptotics and Effective-Rank Barriers for Gaussian Multipole Gram Matrices

This preprint studies large-window effective-depth selection for finite Gaussian multipole Gram matrices. It develops a fixed-dictionary, full-space-normalized framework for window-dependent effective depth, proves fixed-depth large-window convergence, establishes Gaussian-damped tail estimates and finite-range threshold stabilization under separated full-space margins, and reports reproducible Stage A/B finite-grid diagnostics as hypothesis triage rather than asymptotic proof. The manuscript distinguishes theorem-level results from numerical diagnostics through an explicit claim-control framework. Numerical material is generated from the included scripts and configuration files; no external experimental datasets are used.