  We explore BPS quivers for D=5 theories, compactified on a circle and
geometrically engineered over local Calabi-Yau 3-folds, for which many of known
machineries leading to (refined) indices fail due to the fine-tuning of the
superpotential. For Abelian quivers, the counting reduces to a geometric one,
but the technically challenging $L^2$ cohomology proved to be essential for
sensible BPS spectra. We offer a mathematical theorem to remedy the difficulty,
but for non-Abelian quivers, the cohomology approach itself fails because the
relevant wavefunctions are inherently gauge-theoretical. For the Cartan part of
gauge multiplets, which suffers no wall-crossing, we resort to the D0 picture
and reconstruct entire KK towers. We also perform numerical checks using a
multi-center Coulombic routine, with a simple hypothesis on the quiver
invariants, and extend this to electric BPS states in the weak coupling
chamber. We close with a comment on known Donaldson-Thomas invariants and on
how $L^2$ index might be read off from these.
