  We construct infinitely many new exactly solvable local commuting projector
lattice Hamiltonian models for general bosonic beyond group cohomology
invertible topological phases of order two and four in any spacetime
dimensions, whose boundaries are characterized by gravitational anomalies.
Examples include the beyond group cohomology invertible phase without symmetry
in (4+1)D that has an anomalous boundary $\mathbb{Z}_2$ topological order with
fermionic particle and fermionic loop excitations that have mutual $\pi$
statistics. We argue that this construction gives a new non-trivial quantum
cellular automaton (QCA) in (4+1)D of order two. We also present an explicit
construction of gapped symmetric boundary state for the bosonic beyond group
cohomology invertible phase with unitary $\mathbb{Z}_2$ symmetry in (4+1)D. We
discuss new quantum phase transitions protected by different invertible phases
across the transitions.
