Crucial questions about solar and supernova neutrinos remain unanswered. Super-Kamiokande has the exposure needed for progress, but detector backgrounds are a limiting factor. A leading component is the beta decays of isotopes produced by cosmic-ray muons and their secondaries, which initiate nuclear spallation reactions. Cuts of events after and surrounding muon tracks reduce this spallation decay background by $simeq 90%$ (at a cost of $simeq 20%$ deadtime), but its rate at 6--18 MeV is still dominant. A better way to cut this background was suggested in a Super-Kamiokande paper [Bays {it et al.}, Phys.~Rev.~D {bf 85}, 052007 (2012)] on a search for the diffuse supernova neutrino background. They found that spallation decays above 16 MeV were preceded near the same location by a peak in the apparent Cherenkov light profile from the muon; a more aggressive cut was applied to a limited section of the muon track, leading to decreased background without increased deadtime. We put their empirical discovery on a firm theoretical foundation. We show that almost all spallation decay isotopes are produced by muon-induced showers and that these showers are rare enough and energetic enough to be identifiable. This is the first such demonstration for any detector. We detail how the physics of showers explains the peak in the muon Cherenkov light profile and other Super-K observations. Our results provide a physical basis for practical improvements in background rejection that will benefit multiple studies. For solar neutrinos, in particular, it should be possible to dramatically reduce backgrounds at energies as low as 6 MeV.
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