  In this paper, we prove stability or instability of solitons for the
cubic-quintic nonlinear Schrodinger equation at every frequency. The
monotonicity conjecture raised by Killip, Oh, Pocovnicu and Visan is resolved.
We introduce and solve a new cross-constrained variational problem. Then the
uniqueness of the energy minimizers is proposed and shown. According to a
spectral approach and variational arguments, we develop a set of geometric
analysis methods. Correspondences between the soliton frequency and the
prescribed mass are established. Classification of normalized solutions is
given.
