Singleton Optimality in Standard Quadratic Programs with the GOE

We study the standard quadratic optimization problem over the simplex when the objective matrix is drawn from the Gaussian Orthogonal Ensemble (GOE). Let $\kappa_n$ denote the support size of the almost surely unique global optimizer. We prove $$\Prob(\kappa_n>1)\sim 2\sqrt{2\pi}\,\frac{\sqrt{\log n}}{n}.$$ The proof combines an exact two-coordinate condition for edge improvement with a product formula obtained by conditioning on the diagonal order statistics. Boundary-layer estimates identify the leading contribution and show that supports of size at least three are negligible. Consequently, the minimum-diagonal vertex is globally optimal with probability tending to one, with an explicit first-order correction.