  Fr\"olich condensation is a room-temperature nonequilibrium phenomenon which
is expected to occur in many physical and biological systems. Though predicted
theoretically a half century ago, the nature of such condensation remains
elusive. In this Letter, we derive a full quantum theory of Fr\"ohlich
condensation from the Wu-Austin Hamiltonian and present for the first time an
analytical proof that a second-order phase transition induced by nonequilibrium
and nonlinearity emerges in the large-$D$ limit with and without decorrelation
approximation. This critical behavior cannot be witnessed if external sources
are treated classically. We show that the phase transition is accompanied by
large fluctuations in the statistical distribution of condensate phonons and
that the Mandel-Q factor which characterizes fluctuations becomes negative in
the limit of excessive external energy input. In contrast with the cold atom
equilibrium BEC, the Fr\"ohlich condensate is a result of the nonequilibrium
driving where the pump plays a role of setting the number of particles, and the
medium plays a role of setting the temperature. Hence, BEC can either arise by
reducing the medium temperature at fixed pump (equilibrium case), or by
increasing the pump at fixed medium temperature (nonequilibrium case).
