  Celestial amplitudes represent 4D scattering of particles in boost, rather
than the usual energy-momentum, eigenstates and hence are sensitive to both UV
and IR physics. We show that known UV and IR properties of quantum gravity
translate into powerful constraints on the analytic structure of celestial
amplitudes. For example the soft UV behavior of quantum gravity is shown to
imply that the exact four-particle scattering amplitude is meromorphic in the
complex boost weight plane with poles confined to even integers on the negative
real axis. Would-be poles on the positive real axis from UV asymptotics are
shown to be erased by a flat space analog of the AdS resolution of the bulk
point singularity. The residues of the poles on the negative axis are
identified with operator coefficients in the IR effective action. Far along the
real positive axis, the scattering is argued to grow exponentially according to
the black hole area law. Exclusive amplitudes are shown to simply factorize
into conformally hard and conformally soft factors. The soft factor contains
all IR divergences and is given by a celestial current algebra correlator of
Goldstone bosons from spontaneously broken asymptotic symmetries. The hard
factor describes the scattering of hard particles together with the
boost-eigenstate clouds of soft photons or gravitons required by asymptotic
symmetries. These provide an IR safe $\mathcal{S}$-matrix for the scattering of
hard particles.
