  We prove that every unitary two-dimensional conformal field theory (with no
extended chiral algebra, and with central charges c_L, c_R > 1) contains a
primary operator with dimension Delta_1 that satisfies 0 < Delta_1 < (c_L +
c_R)/12 + 0.473695. Translated into gravitational language using the AdS_3
/CFT_2 dictionary, our result proves rigorously that the lightest massive
excitation in any theory of 3D gravity with cosmological constant Lambda < 0
can be no heavier than 1/(4 G_N) + o(|Lambda|^(1/2)). In the flat-space
approximation, this limiting mass is twice that of the lightest BTZ black hole.
The derivation of the bound applies at finite central charge for the CFT, and
does not rely on an asymptotic expansion at large central charge. Neither does
our proof rely on any special property of the CFT such as supersymmetry or
holomorphic factorization, nor on any bulk interpretation in terms of string
theory or semiclassical gravity. Our only assumptions are unitarity and modular
invariance of the dual CFT. Our proof demonstrates for the first time that
there exists a universal center-of-mass energy beyond which a theory of "pure"
quantum gravity can never consistently be extended.
