  We propose a high statistics experiment to search for invisible decay modes
in nuclear gamma cascades. A radioactive source (such as $^{60}$Co or
$^{24}$Na) that triggers gamma cascades is placed in the middle of a large,
hermetically sealed scintillation detector, enabling photon identification with
high accuracy. Invisible modes are identified by establishing the absence of a
photon in a well-identified gamma cascade. We propose the use of fast
scintillators with nanosecond timing resolution, permitting event rates as high
as $10^{7}$ Hz. Our analysis of the feasibility of this setup indicates that
branching fractions as small as $10^{-12} - 10^{-14}$ can be probed. This
experimental protocol benefits from the fact that a search for invisible modes
is penalized for weak coupling only in the production of the new particle. If
successfully implemented, this experiment is an exquisite probe of particles
with mass below $\sim$4 MeV that lie in the poorly constrained supernova
"trapping window" that exists between 100 keV - 30 MeV. Such particles have
been invoked as mediators between dark matter and nucleons, explain the proton
radius and $(g-2)_{\mu}$ anomalies and potentially power the shock wave in type
II supernovae. The hadronic axion could also be probed with modifications to
the proposed setup.
