Isolations of cubic lattices from their proper sublattices (Q6135865)

A positive definite integral quadratic form is called an isolation of a quadratic form \(f\) if it represents all subforms of \(f\) except for \(f\) itself. The minimum rank of isolations of a quadratic form \(f\), if it exists, is denoted by Iso(\(f\)). It is noted here that the existence of an isolation of a form \(f\) is equivalent to the set \(\{f\}\) being the unique minimal \(S_f\)-universality criterion set, where \(S_f\) denotes the set of all subforms of \(f\). In this paper, the author considers isolations of the form \(I_n\) corresponding to the sum of \(n\) squares. It is proven that such an isolation exists for every positive integer \(n\). The specific values \(\mathrm{Iso}(I_2)=5\) and \(\mathrm{Iso}(I_3)=6\) are determined, and explicit lower and upper bounds are given for \(\mathrm{Iso}(I_n)\) for any positive integer \(n\).