Two-Centre Multipole Gram Certification in Structured Gaussian Traps: Block Stability, Effective-Depth Obstructions, and Nonseparable Stress Tests

This preprint studies finite-depth multipole Gram certification for a two-centre structured Gaussian trap. It combines a finite-dimensional block-stability theorem with reproducible numerical diagnostics up to depth L=3, threshold-sensitivity tests, and weak nonseparable perturbation stress tests. The manuscript distinguishes numerical threshold validity from theorem-level block certification and documents effective-depth obstructions arising from inter-centre leakage and dictionary conditioning.