Why are crystalline structures in solids so common? Is there any sort of theorem or paper that shows that periodic arrays gives ground states? Or any other theoretical reason why crystals are so common?
 A: (Since several weeks have passed without the requested rigorous proof, I'll give the hand-wavy reasoning that came to mind.)
Given that certain bond angles are favorable, it makes intuitive sense to aim to reproduce them identically at every atom. This is equivalent to periodicity.
Even if certain bond angles aren't favorable, periodic stacking (e.g., in face-centered-cubic crystals) still puts each atom as close as it would like to be to its neighbor, which is ideal.
However, this reasoning—enthalpic minimization—doesn't give the whole picture. At constant temperature and pressure, it's not the enthalpy $H$ but the Gibbs free energy $G\equiv H-TS$ that's minimized, as also affected by the temperature $T$ and entropy $S$. A higher-entropy that sacrifices some bonding benefits might be favored, especially at higher temperatures. This is why everything becomes a gas upon sufficient heating, even though a periodic solid features excellent bonding.
Furthermore, the periodic state may be thermodynamically favorable but kinetically limited; this is the general reason why we see amorphous solids around us. These so-called glasses want to be crystals.
