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.
