  Let $Z_1,\ldots,Z_n$ be i.i.d. isotropic random vectors in $\mathbb{R}^p$,
and $T \subset \mathbb{R}^p$ be a compact set. A classical line of empirical
process theory characterizes the size of the suprema of the quadratic process
\begin{align*} \sup_{t \in T} \bigg| \frac{1}{n}\sum_{i=1}^n \langle Z_i,t
\rangle^2-\lVert t \rVert^2 \bigg|, \end{align*} via a single parameter known
as the Gaussian width of $T$.
  This paper introduces an improved bound for the suprema of this quadratic
process for standard Gaussian vectors $\{Z_i\}$ that can be exactly attained
for certain choices of $T$, and is thus referred to as an exact bound. Our
exact bound is expressed via a collection of (stochastic) Gaussian widths over
spherical sections of $T$ that serves as a natural multi-scale analogue to the
Gaussian width of $T$. Compared to the classical bounds for the quadratic
process, our new bounds not only determine the optimal constants in the
classical bounds that can be attained for some $T$, but also precisely capture
certain subtle phase transitional behavior of the quadratic process beyond the
reach of the classical bounds.
  To illustrate the utility of our results, we obtain tight versions of the
Gaussian Dvoretzky-Milman theorem for random projection, and the
Koltchinskii-Lounici theorem for covariance estimation, both with optimal
constants. Moreover, our bounds recover the celebrated BBP phase transitional
behavior of the top eigenvalue of the sample covariance and its generalization
to the sample covariance error.
  The proof of our results exploits recently sharpened Gaussian comparison
inequalities. The technical scope of our method of proof is further
demonstrated in obtaining an exact bound for a two-sided Chevet inequality.
