  Combining insights from both the effective field theory of quantum gravity
and black hole thermodynamics, we derive two novel consistency relations to be
satisfied by any quantum theory of gravity. First, we show that a particular
combination of the number of massless (light) fields in the theory must take
integer values. Second, we show that, once the massless spectrum is fixed, the
Wilson coefficient of the Kretschmann scalar in the low-energy effective theory
is fully determined by the logarithm of a single natural number.
