Explicit Separators for Consecutive Levels of Parrilo's Sum-of-Squares Hierarchy over the Copositive Cone

Abstract

Parrilo's cones $\Kc{n}{r}$ form a nested sequence of semidefinite-representable inner approximations of the copositive cone $\COP_n$. For $n=5$ their union is all of $\COP_5$, yet no single level attains it, and whether consecutive levels actually differ had remained open beyond the classical first step. No explicit matrix in $\Kc{n}{t}\setminus\Kc{n}{t-1}$ had, to our knowledge, been published for any $t\ge2$ and $n\ge5$. We settle the first three cases. Explicit rational matrices, obtained from diagonal scalings of the Horn matrix shifted along a positive interior direction, lie in $\Kc{5}{2}\setminus\Kc{5}{1}$, in $\Kc{5}{3}\setminus\Kc{5}{2}$, and in $\Kc{5}{4}\setminus\Kc{5}{3}$, giving three consecutive strict inclusions $\Kc{5}{1}\subsetneq\Kc{5}{2}\subsetneq\Kc{5}{3}\subsetneq\Kc{5}{4}$. Each is certified by an exact rational Gram matrix and an exact rational dual moment functional, re-verified by a standalone program in integer arithmetic. The separations are robust. One fixed certificate pair covers an interval of shifts of width exceeding $3\cdot10^{-3}$, and $\Kc{5}{2}\setminus\Kc{5}{1}$ has nonempty interior. Combining a scaling theorem of Dickinson, Dür, Gijben and Hildebrand with the completeness theorem of Schweighofer and Vargas shows further that strict adjacent inclusions recur at arbitrarily large levels. All separators were located by one threshold device: the least shift $\eps_r(M)$ carrying $M$ into $\Kc{5}{r}$ along an interior direction is nonincreasing in $r$, and each strict drop between levels marks a window of separators.