Trap-Weighted Multipole Gram Certification: Finite-Depth Riesz Stability and Selection Rules

This preprint develops a bounded-window trap-weighted multipole Gram certification framework for finite-depth Riesz stability. The manuscript focuses on radial harmonic traps and weak radial-angular anisotropic perturbations on bounded observation windows B_R. The main components are a normalized baseline Gram construction, an exact radial block certificate, Gaunt/Wigner selection rules for angular couplings, and a finite-depth weak-anisotropy certificate with an explicit spectral-norm remainder. The record includes the manuscript PDF and a reproducibility/source package containing the LaTeX source, benchmark script, numerical tables, figures, and README. No external experimental datasets are used; the numerical material consists of manufactured diagnostic computations generated from the equations and parameters stated in the manuscript.