Gaussian Gram matrix concentration on a fixed subspace ($P(\Vert G^{T}G/n-I_k{\Vert }_{\mathrm{op}}>1/2)\le 2e^{k\log B-n/1024}$)

For $G$ with $n$ independent standard Gaussian rows in dimension $k$, combine a constant-radius sphere metric net, the quadratic form net bound, the chi-squared concentration inequality, and a union bound. This gives $P(\Vert G^{T}G/n-I_k{\Vert }_{\mathrm{op}}>1/2)\le 2\exp (k\log B-n/1024)$ for a numerical $B$. Taking $n$ larger than a constant times $k\log p$ gives exponential decay in $k\log p$. The subspace is fixed, so no union over supports is needed.