  The Sun is a bright source of GeV gamma rays, due to cosmic rays interacting
with solar matter and photons. Key aspects of the underlying processes remain
mysterious. The emission in the TeV range, for which there are neither
observational nor theoretical studies, could provide crucial clues. The new
experiments HAWC (running) and LHAASO (planned) can look at the Sun with
unprecedented sensitivity. In this paper, we predict the very high-energy (up
to 1000 TeV) gamma-ray flux from the solar disk and halo, due to cosmic-ray
hadrons and electrons ($e^++e^-$), respectively. We neglect solar magnetic
effects, which is valid at TeV energies; at lower energies, this gives a
theoretical lower bound on the disk flux and a theoretical upper bound on the
halo flux. We show that the solar-halo gamma-ray flux allows the first test of
the $\sim 5$--70 TeV cosmic-ray electron spectrum. Further, we show that HAWC
can immediately make an even stronger test with nondirectional observations of
cosmic-ray electrons. Together, these gamma-ray and electron studies will
provide new insights about the local density of cosmic rays and their
interactions with the Sun and its magnetic environment. These studies will also
be an important input to tests of new physics, including dark matter.
