  We investigate the hypothesis of dynamic tetragonal domains occuring in cubic
CH$_3$NH$_3$PbI$_3$ using high-resolution neutron spectroscopy to study a fully
deuterated single crystal. The $\textit{R}$-point scattering above the 327.5 K
tetragonal-cubic phase transition is always resolution-limited in energy and
therefore inconsistent with dynamic domain predictions. This behavior is
instead consistent with the central peak phenomenon observed in other
perovskites. The scattering may originate from small, static, tetragonal-phase
domains nucleating about crystal defects, and the temperature dependence
demonstrates the transition is first-order.
