  We generalize the existing finite-size criteria for spectral gaps of
frustration-free spin systems to $D>2$ dimensions. We obtain a local gap
threshold of $\frac{3}{n}$, independent of $D$, for nearest-neighbor
interactions. The $\frac{1}{n}$ scaling persists for arbitrary finite-range
interactions in $\mathbb Z^3$. The key observation is that there is more
flexibility in Knabe's combinatorial approach if one employs the operator
Cauchy-Schwarz inequality.
