 Gaussian hyperplane rounding

Given a real unit-vector Gram matrix $Y_{ij}=⟨v_i,v_j⟩$, draw a standard Gaussian random vector $Z$ and set $y_i=\mathrm{sign}⟨v_i,Z⟩$. Each coordinate is almost surely a sign; zero projections have probability zero. The same random separating hyperplane is used for every coordinate, so the signs need not be independent. Their pair expectations follow the Gaussian sign-correlation identity.