  We prove that the dimension of the intersection $\mathcal Z$ of all Hassett
divisors of special cubic fourfolds is sixteen. We do this by studying which
subsets of the natural numbers $\mathbb N$ can be obtained as the image of a
positive-definite integral quadratic form and what the minimal possible rank of
such a form is. In particular, for the subset of $\mathbb N$ consisting of all
possible discriminants of special cubic fourfolds, we show this rank is four
and that this is the codimension of $\mathcal Z$ in $\mathcal C$, the
twenty-dimensional moduli space of cubic fourfolds.
