It makes sense that particles with certain proper masses are stable while all other values of proper mass are unstable. This does NOT imply the existence of a smallest unit of proper mass, of which heavier particles must consist.

If a chaotic system is close to an attractor, it tends to stabilize into the attractive state. Personally, I believe the stable particles are chaotic attractors in a chaotic soup of regular energy and dark energy. If they have slightly too much energy, they shed just enough energy to reach the nearest stable state.