  We present a simple and versatile method for deriving (an)isotropic local
laws for general random matrices constructed from independent random variables.
Our method is applicable to mean-field random matrices, where all independent
variables have comparable variances. It is entirely insensitive to the
expectation of the matrix. In this paper we focus on the probabilistic part of
the proof -- the derivation of the self-consistent equations. As a concrete
application, we settle in complete generality the local law for Wigner matrices
with arbitrary expectation.
