Bounded-Window Riesz Stability and Effective-Rank Barriers for Anisotropic Multipole Gram Matrices

This preprint studies normalized multipole Gram matrices generated by anisotropic trap weights on a bounded spatial window. Under a positive bounded multiplier assumption on the weight ratio, the normalized Gram matrices are finite-dimensional compressions of a positive bounded multiplication operator. Consequently, at fixed bounded window, their spectra remain enclosed between the essential lower and upper bounds of the multiplier, so true infinite-depth Riesz collapse cannot occur. The paper separates bounded-window Riesz stability from near-identity certification and introduces effective-rank diagnostics for practical finite-depth conditioning. A validated monomial-profile numerical benchmark is included, together with manifest-based provenance and replay-validation files.