  We study bosonic systems on a spacetime lattice defined by path integrals of
commuting fields. We introduce branch-independent bosonic (BIB) systems, whose
path integral is independent of the branch structure of the spacetime
simplicial complex, even for a spacetime with boundaries. In contrast, a
generic lattice bosonic (GLB) system's path integral may depend on the branch
structure. We find the invertible topological order characterized by the
Stiefel-Whitney cocycle (e.g., 4+1d w$_2$w$_3$) to be nontrivial for BIB
systems, but this topological order and a trivial gapped tensor product state
belong to the same phase for GLB systems. The invertible topological orders in
GLB systems are not classified by the oriented cobordism. The branch dependence
on a lattice may be related to the orthonormal frame of smooth manifolds and
the framing anomaly of continuum field theories. The branch structure on a
discretized lattice may be related to a frame structure on a smooth manifold
that trivializes any Stiefel-Whitney classes. We construct BIB systems to
realize the w$_2$w$_3$ topological order, and its 3+1d gapped or gapless
boundaries. A 3+1d $\mathbb{Z}_2$ gauge theory with (1) fermionic
$\mathbb{Z}_2$ gauge charge particle trivializes w$_2$ and (2) fermionic
$\mathbb{Z}_2$ gauge flux line trivializes w$_3$. In particular, if the flux
loop's worldsheet is unorientable, then an orientation-reversal 1d worldline
corresponds to a fermion worldline carrying no $\mathbb{Z}_2$ gauge charge.
Spin and Spin$^c$ structures trivialize the w$_2$w$_3$ global pure
gravitational anomaly to zero (which helps to construct 3+1d $\mathbb{Z}_2$ and
all-fermion U(1) gauge theories), but the Spin$^h$ and
Spin$\times_{\mathbb{Z}_2}$Spin$(n \geq 3)$ structures modify the w$_2$w$_3$
into a global mixed gauge-gravitational anomaly, which helps to constrain Grand
Unifications (e.g., $n=10,18$) or construct new models.
