We define a general class of "crystallographic" sphere packings, and study the subclass of "superintegral" such. We prove an "arithmeticity" theorem, connecting these to the theory of hyperbolic arithmetic reflection lattices. As a consequence, superintegral packings only exist in finitely many dimensions, and in fact in finitely many commensurability classes, in principle allowing for a complete classification.
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/3444
[3] https://www.mpim-bonn.mpg.de/node/7800