... of Finite Non-Zero Mass and Non-Zero Temperature
[Note: I had another name in mind, but, on checking my terms, it seems that "perfect crystal" and "ideal crystal" both refer to crystals without imperfections, rather than perfectly rigid ones.]
[Note: I've had this idea in my head before, but I've never gotten to writing it down before.]
We consider the possibility of the existence of a real object with zero flexibility (or infinite rigidity). However, rather than using math and formulae, we use logic and the overall concepts that have been known in physics for some time.
Suppose that it is possible to construct a perfectly inflexible crystal one lightyear in length and symmetrical about the plane perpendicular to this axis and bisecting its center point (such as a cylinder). Note that, in this case, we are ignoring the thermodynamic difficulties in removing all heat from an object - we soon won't need them. Supposing, then, that we set up monitoring stations on either end of the hypothetical object, and assuming that it will tend to pivot about its center of gravity, any movement applied to one end of the object would be immediately detectable (though in the opposite direction) at the other end. This would entail the travel of information one lightyear in zero time, in violation of relativity. Note that, even if we somehow hold it at the far end, the movement then becomes detectable half of one lightyear away in zero time, which poses the same problem. (Technically, the movement is detectable in many other locations (i.e., all or nearly all points along the structure), but these ones are a bit more illustrative.)
Consider, though, that any non-zero distance poses a problem, since the speed of information travel would still be infinite. Thus, even a perfectly inflexible crystal only three atoms in length would create a contradiction against relativity.
Note, though, that the argument outlined here applies only if there is energy in the system of the perfectly inflexible crystal, such as that applied by the movement of one end. However, completely removing energy from the system is a difficult prospect, as it may entail, among other things, removing gravity from the universe.