Universal Multi-Shell Spectral Locking in Bd and Spectral Resonance in Critical Dimensions (An Eigenvalue-Based PSD Barrier Forcing Stray = 0)

This manuscript presents a rigorous analytical proof of "Harmonic Rigidity" and universal spectral locking in $B_d$-symmetric multi-shell point configurations. By employing second-order eigenvalue perturbation theory on the rank-deficient Gram manifold, we prove that any geometric deviation (stray drift) from the exact Cohn-Elkies arithmetic spectrum induces a strictly negative smallest eigenvalue ($\lambda_{min}^{(2)} < 0$) in active principal minors. This establishes an absolute Positive Semidefinite (PSD) barrier, mathematically forcing the $\mathrm{Stray}=0$ spectral resonance condition. Furthermore, we formally prove that this infinitesimal rigidity is irreducible to classical Connelly-Gortler-Thurston (CGT) equivariant stress matrices, demonstrating that convergence to optimal lattice roots ($E_8$, Leech) is an analytical necessity rather than a mere numerical observation.