  We show that, if a rational homology 3-sphere $Y$ bounds a positive definite
smooth 4-manifold, then there are finitely many negative definite lattices, up
to the stable-equivalence, which can be realized as the intersection form of a
smooth 4-manifold bounded by $Y$. To this end, we make use of constraints on
definite forms bounded by $Y$ induced from Donaldson's diagonalization theorem,
and correction term invariants due to Fr\o yshov, and Ozsv\'ath and Szab\'o. In
particular, we prove that all spherical 3-manifolds satisfy such finiteness
property.
