  We examine in detail the higher spin fields which arise on the basic fuzzy
sphere $S^4_N$ in the semi-classical limit. The space of functions can be
identified with functions on classical $S^4$ taking values in a higher spin
algebra associated to $\mathfrak{so}(5)$. We derive an explicit and complete
classification of the scalars and one-forms on the semi-classical limit of
$S_N^4$. The resulting kinematics is reminiscent of Vasiliev theory. Yang-Mills
matrix models naturally provide an action formulation for higher spin gauge
theory on $S^4$, with 4 irreducible modes for each spin $s\geq 1$. We
diagonalize the quadratic part of the effective action and exactly evaluate the
quadratic part in the spin 2 sector. By identifying the linear perturbation of
the effective metric, we obtain the exact kinetic term for all graviton
candidates. At the classical level, matter $T_{\mu\nu}$ leads to three
different contributions to the linearized metric: one consistent with
linearized GR, one more rapidly decreasing contribution, and one
non-propagating contribution localized at $T_{\mu\nu}$. The latter is too large
to be physically acceptable, unless there is a significant induced quantum
action. This issue should be resolved on generalized fuzzy spaces.
