Why are (most) solids periodic? Is there a rigorous proof that periodic arrangements minimize the energy of a group of particles?
 A: I have the feeling that there may not be a rigourous proof of the type you describe in your question. 
I remember a couple of years ago there was a proof that the close packed structures (ccp / hcp) gave the best 'space filling' characteristics. This proof by Hales was confirmed in August of 2014 The proof relies in part on computer checking that other possibilites have lower space filling ratios. Space filling is a measure of how much the space in a volume of a lattice is filled by spheres of fixed radius. 
Given that close packed structures have only just proved to be the most efficient way of packing spheres I doubt the rigorous proof you are asking about has been produced. 
I would also point out that your question is quite general and does not consider the binding mechanism between particles. In some cases, metals primarily, there may be little of no directionality preferred and the lowest energy structures are most closely packed.
In other solids, such as diamond, the shape of the solid reflects the very directional nature of the links between different carbon atoms.  
