The dark dimension, proton decay, and the length of the M-theory interval

The existence of a large extra dimension in which only gravity propagates would have spectacular consequences for cosmology and laboratory experiments. In the strong coupling limit of the $E_8\times E_8$ heterotic string theory, the gauge and matter fields live at the end of the eleventh dimension, which becomes a natural candidate for a micron-size $\mathit{\text{dark}}\mathit{\text{dimension}}$. In this work, however, we show that the length of the M-theory interval is severely constrained by proton decay searches. Our results indicate that in such constructions the size of the eleventh dimension is $R\lesssim \mathcal{O}({10}^{-28})$ meters.