  We prove a general local law for Wigner matrices which optimally handles
observables of arbitrary rank and thus it unifies the well-known averaged and
isotropic local laws. As an application, we prove that the quadratic forms of a
general deterministic matrix $A$ on the bulk eigenvectors of a Wigner matrix
has approximately Gaussian fluctuation. For the bulk spectrum, we thus
generalize our previous result [arXiv:2103.06730] valid for test matrices $A$
of large rank as well as the result of Benigni and Lopatto [arXiv:2103.12013]
valid for specific small rank observables.
