  The bulk-boundary correspondence, a topic of intensive research interest over
the past decades, is one of the quintessential ideas in the physics of
topological quantum matter. Nevertheless, it has not been proven in all
generality and has in certain scenarios even been shown to fail, depending on
the boundary profiles of the terminated system. Here, we introduce bulk numbers
that capture the exact number of in-gap modes, without any such subtleties in
one spatial dimension. Similarly, based on these 1D bulk numbers, we define a
new 2D winding number, which we call the pole winding number, that specifies
the number of robust metallic surface bands in the gap as well as their
topological character. The underlying general methodology relies on a simple
continuous extrapolation from the bulk to the boundary, while tracking the
evolution of Green's function's poles in the vicinity of the bulk band edges.
As a main result we find that all the obtained numbers can be applied to the
known insulating phases in a unified manner regardless of the specific
symmetries. Additionally, from a computational point of view, these numbers can
be effectively evaluated without any gauge fixing problems. In particular, we
directly apply our bulk-boundary correspondence construction to various
systems, including 1D examples without a traditional bulk-boundary
correspondence, and predict the existence of boundary modes on various
experimentally studied graphene edges, such as open boundaries and grain
boundaries. Finally, we sketch the 3D generalization of the pole winding number
by in the context of topological insulators.
