  We study the eigenvector mass distribution for generalized Wigner matrices on
a set of coordinates $I$, where $N^\varepsilon \le | I | \le N^{1-
\varepsilon}$, and prove it converges to a Gaussian at every energy level,
including the edge, as $N\rightarrow \infty$. This result represents a strong
from of Quantum Unique Ergodicity for random matrices. The key technical input
is a four-point decorrelation estimate for eigenvectors of matrices with a
large Gaussian component. Its proof is an application of the maximum principle
to a new set of moment observables satisfying parabolic evolution equations.
