I'm confused. Protons are "splittable" in the sense that they are not elementary particles, but are instead comprised of multiple quarks. The Large Hadron Collider [0] "splits" protons. In fact this is how they claim to have observed the Higgs boson. See the image caption:
Simulated Large Hadron Collider CMS particle
detector data depicting a Higgs boson produced
by colliding protons decaying into hadron jets
and electrons
