  The existence and stability of atoms rely on the fact that neutrons are more
massive than protons. The measured mass difference is only 0.14\% of the
average of the two masses. A slightly smaller or larger value would have led to
a dramatically different universe. Here, we show that this difference results
from the competition between electromagnetic and mass isospin breaking effects.
We performed lattice quantum-chromodynamics and quantum-electrodynamics
computations with four nondegenerate Wilson fermion flavors and computed the
neutron-proton mass-splitting with an accuracy of $300$ kilo-electron volts,
which is greater than $0$ by $5$ standard deviations. We also determine the
splittings in the $\Sigma$, $\Xi$, $D$ and $\Xi_{cc}$ isospin multiplets,
exceeding in some cases the precision of experimental measurements.
