  We prove using invariance under the modular $S$- and $ST$-transformations
that every unitary two-dimensional conformal field theory (CFT) of only
even-spin operators (with no extended chiral algebra and with central charges
$c,\tilde{c}>1$) contains a primary operator with dimension $\Delta_1$
satisfying $0 < \Delta_1 < (c+\tilde{c})/24 + 0.09280...$ After deriving both
analytical and numerical bounds, we discuss how to extend our methods to bound
higher conformal dimensions before deriving lower and upper bounds on the
number of primary operators in a given energy range. Using the AdS$_3$/CFT$_2$
dictionary, the bound on $\Delta_1$ proves the lightest massive excitation in
appropriate theories of 3D matter and gravity with cosmological constant
$\Lambda < 0$ can be no heavier than $1/(8G_N)+O(\sqrt{-\Lambda})$; the bounds
on the number operators are related via AdS/CFT to the entropy of states in the
dual gravitational theory. In the flat-space approximation, the limiting mass
is exactly that of the lightest BTZ black hole.
