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We construct an exactly solvable commuting projector model for a $4+1$dimensional ${\mathbb Z}_2$ symmetry-protected topological phase (SPT) which isoutside the cohomology classification of SPTs. The model is described by adecorated domain wall construction, with "three-fermion" Walker-Wang phases onthe domain walls. We describe the anomalous nature of the phase in severalways. One interesting feature is that, in contrast to in-cohomology phases, theeffective ${\mathbb Z}_2$ symmetry on a $3+1$ dimensional boundary cannot bedescribed by a quantum circuit and instead is a nontrivial quantum cellularautomaton (QCA). A related property is that a codimension-two defect (forexample, the termination of a ${\mathbb Z}_2$ domain wall at a trivialboundary) will carry nontrivial chiral central charge $4$ mod $8$. We alsoconstruct a gapped symmetric topologically-ordered boundary state for ourmodel, which constitutes an anomalous symmetry enriched topological phaseoutside of the classification of arXiv:1602.00187, and define a correspondinganomaly indicator.

An Exactly Solvable Model for a $4+1D$ Beyond-Cohomology Symmetry Protected Topological Phase