  We use the method of interlacing families of polynomials introduced to prove
two theorems known to imply a positive solution to the Kadison--Singer problem.
  The first is Weaver's conjecture $KS_{2}$ \cite{weaver}, which is known to
imply Kadison--Singer via a projection paving conjecture of Akemann and
Anderson. The second is a formulation due to Casazza, et al., of Anderson's
original paving conjecture(s), for which we are able to compute explicit paving
bounds.
  The proof involves an analysis of the largest roots of a family of
polynomials that we call the "mixed characteristic polynomials" of a collection
of matrices.
