  Quantum gravitational effects may induce stochastic fluctuations in the
structure of space-time, to produce a characteristic foamy structure. It has
been known for some time now that these fluctuations may have observable
consequencies for the propagation of cosmic ray particles over cosmological
distances. We note here that the same fluctuations, if they exist, imply that
some decay reactions normally forbidden by elementary conservation laws, become
kinematically allowed, inducing the decay of particles that are seen to be
stable in our universe. Due to the strength of the prediction, we are led to
consider this finding as the most severe constraint on the classes of models
that may describe the effects of gravity on the structure of space-time. We
also propose and discuss several potential loopholes of our approach, that may
affect our conclusions. In particular, we try to identify the situations in
which despite a fluctuating energy-momentum of the particles, the reactions
mentioned above may not take place.
