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In this work we study gapped boundary states of $\mathbb{Z}_N$ bosonicsymmetry-protected topological (SPT) phases in 4+1d, which are characterized bymixed $\mathbb{Z}_N$-gravity response, and the closely related phases protectedby $C_N$ rotation symmetry. We show that if $N\notin \{2,4,8,16\}$, anysymmetry-preserving boundary theory is necessarily gapless for the root SPTstate. We then propose a 3+1d $\mathbb{Z}_2$ gauge theory coupled to fermionicmatter as a candidate boundary theory for $N=2,4,8,16$, where the anomaloussymmetry is implemented by invertible topological defects obtained from gauging2+1d chiral topological superconductors. For the $C_N$ case, we present anexplicit construction for the boundary states for $N=2,4,8,16$, and argue thatthe construction fails for other values of $N$.

Gapped boundary of (4+1)d beyond-cohomology bosonic SPT phase