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The holographic duality gives a one-to-one mapping between consistent theories of gravity with AdS_d ground state, and conformal field theories in d-1 dimensions. Since the latter are unambiguously well-defined, we can use their defining properties to prove rigorous, model-independent theorems covering all quantum gravity theories with negative cosmological constant. Taking d=3, I will prove in this talk that every two-dimensional CFT contains a primary perator of weight less than C /12 + 0.473695, where C is the sum of right- and left-moving central charges. Using the AdS/CFT dictionary, this establishes that every theory of quantum gravity in asymptotically AdS space contains some state of mass less than 1 / (4 G), above and beyond the universal sector of boundary metric excitations.