proton decay
The proton sitting in the heart of every hydrogen atom is, as far as we have ever seen, completely permanent. Electrons, photons and protons are the rocks the rest of matter is built on; if protons fell apart, atoms themselves would have a shelf life. Proton decay is the predicted but never-observed process in which a proton finally breaks down into lighter particles — for example a positron and a neutral pion — meaning ordinary matter is not eternal after all, just astonishingly long-lived.
In the Standard Model the proton is protected by a bookkeeping rule called baryon-number conservation, which simply forbids it from decaying. But grand unified theories, by placing quarks and leptons in the same family, naturally allow a quark to turn into a lepton, and that quietly breaks the protection. The catch is the timescale: the predicted proton lifetime is unimaginably long — at least around 10^34 years, which is a trillion trillion times the current age of the universe. You cannot wait that long, so instead you watch an enormous number of protons at once: stare at a huge tank of water and even a faint decay rate means a few should pop within a year.
Proton decay matters because it is the cleanest laboratory test of grand unification we have. Detectors holding tens of thousands of tonnes of ultrapure water, deep underground to shield out cosmic rays, have watched for decades and seen not one unambiguous proton death. Those non-detections have already excluded the simplest grand unified theory and keep tightening the screws on the rest. It remains one of physics' great patient experiments: a discovery would be revolutionary, and even the continuing silence is informative.
Super-Kamiokande, a tank holding fifty thousand tonnes of water lined with light sensors, watches for the single bright ring a positron would leave if a proton ever decayed; after decades it has set the proton lifetime at more than 10^34 years.
Watching millions of trillions of protons at once to catch one rare death.
Proton decay has never been observed; every number quoted is a lower limit on the lifetime, not a measurement of it. The proton may simply be stable, in which case the grand unified theories that require its decay are wrong.