  In this work we study gapped boundary states of $\mathbb{Z}_N$ bosonic
symmetry-protected topological (SPT) phases in 4+1d, which are characterized by
mixed $\mathbb{Z}_N$-gravity response, and the closely related phases protected
by $C_N$ rotation symmetry. We show that if $N\notin \{2,4,8,16\}$, any
symmetry-preserving boundary theory is necessarily gapless for the root SPT
state. We then propose a (3+1)$d$ $\mathbb{Z}_2$ gauge theory coupled to
fermionic matter as a candidate boundary theory for $N=2,4,8,16$, where the
anomalous symmetry is implemented by invertible topological defects obtained
from gauging (2+1)$d$ chiral topological superconductors. For the $C_N$ case,
we present an explicit construction for the boundary states for $N=2,4,8,16$,
and argue that the construction fails for other values of $N$.
