  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.
