  Motivated by the Pati-Salam Grand Unified Theory, we study $(4+1)d$
topological insulators with $SU(4) \times SU(2)_1 \times SU(2)_2$ symmetry,
whose $(3+1)d$ boundary has 16 flavors of left-chiral fermions, which form
representations $(\mathbf{4}, \mathbf{2}, \mathbf{1})$ and $(\bar{\mathbf{4}},
\mathbf{1}, \mathbf{2})$. The key result we obtain is that, without any
interaction, this topological insulator has a $\mathbb{Z}$ classification,
namely any quadratic fermion mass operator at the $(3+1)d $ boundary is
prohibited by the symmetries listed above; while under interaction this system
becomes trivial, namely its $(3+1)d$ boundary can be gapped out by a properly
designed short range interaction without generating nonzero vacuum expectation
value of any fermion bilinear mass, or in other words, its $(3+1)d$ boundary
can be driven into a "strongly coupled symmetric gapped (SCSG) phase". Based on
this observation, we propose that after coupling the system to a dynamical
$SU(4) \times SU(2)_1 \times SU(2)_2$ lattice gauge field, the Pati-Salam GUT
can be fully regularized as the boundary states of a $(4+1)d$ topological
insulator with a {\it thin} fourth spatial dimension, the thin fourth dimension
makes the entire system generically a $(3+1)d$ system. The mirror sector on the
opposite boundary will {\it not} interfere with the desired GUT, because the
mirror sector is driven to the SCSG phase by a carefully designed interaction
and is hence decoupled from the GUT.
