Spectral Galerkin methods are renowned for high-precision eigenvalue approximation, yet a rigorous lower bound obtained directly from a spectral discretization has remained unavailable: the classical Kato and Weinstein–Temple enclosures do apply but require a priori information on a neighboring eigenvalue. This research resolves the issue by extending the speaker’s projection-based framework for guaranteed lower eigenvalue bounds—so far realized only through finite element methods—to conforming spectral Galerkin methods. For $-\Delta+V$ with $0\le V \in L^\infty$ , a projection-gap estimate yields an explicit constant for the standard Galerkin matrix (exact at $V=0$ ), and a composite discretization removes the $\|V\|_{\infty}$-dependence for large potentials. The same auxiliary-projector mechanism extends to singular potentials with an unbounded $L^\infty$ norm—in particular to attractive Coulomb singularities in three dimensions, via a localized Hardy inequality—which we develop in near future.