Ok, so the “Born–von Karman boundary condition” is helpful in understanding the center area of the brownie. It’s homogenous which is why you can assume the “band index” is periodic / regular and countable. It has to be infinite to work. So, no edges. No edges to the brownie. As I’m interested in the edges, I know where not to look and where something I’m looking at is in the center of the brownie.
There’s some important assumptions.
https://en.wikipedia.org/wiki/Electronic_band_structure
“Modeling the potential of a crystal as a periodic function with the Born–von Karman boundary condition and plugging in Schrödinger’s equation results in a proof of Bloch’s theorem, which is particularly important in understanding the band structure of crystals.”