We introduce quasinormality and pseudonormality for nonlinear second-order cone programming by exploiting the spectral geometry of the Lorentz cone. The proposed quasinormality distinguishes interior and boundary multipliers and, in the latter case, relates constraint violation to the corresponding spectral direction. It recovers classical quasinormality when all cone blocks are one-dimensional and is strictly weaker than Robinson’s condition. Nevertheless, it implies a local error bound and exact penalization, while also guaranteeing boundedness of dual subsequences generated by an augmented Lagrangian method at feasible accumulation points. We further introduce a relaxed formulation, expressed solely in terms of projections, norms, and distances to the cone boundary, that preserves the local error bound and dual boundedness. Finally, an extension of this condition to infeasible points allows us to establish feasibility of accumulation points satisfying the extended condition.