  We introduce an unconventional regularization of condensed matter effective
field theory by a non-local bulk topological regulator in one higher dimension,
which is equivalent to a bulk-interface-bulk formulation. We show its necessity
motivated by a (0+1)-dimensional system. Although the system can be strictly
defined on the lattice in its own dimensions, the bulk regulator is important
to derive its correct physical observables, which cannot be captured by any
local regulator. We explicitly obtain a constraint on the thermal and electric
integer Hall conductances of half-filled translation invariant $N$-flavor
gapped fermionic system on a square lattice possessing a unique ground state
with uniform rational magnetic fluxes per unit cell in the presence of the
onsite $U(N)$ symmetry. The Wiedemann-Franz law is shown to be obeyed by the
Hall conductances regardless of arbitrarily strong interactions. We further
obtain no-go theorems on the possible symmetric gapped phases.
