[raattgift]

> how do we know the periodicity is there ... Just theory?

Theory guides us, but experimentally you can for example make a whole bunch of time crystals (especially straightfoward for driven time crystals, which have a period that's an integer multiple of the driver, the driving force being laser light or microwaves) with an expected set of states it cycles through, and you can test those states once per time crystal. If you reliably get the states theory predicts, that's good evidence.

> could you outline the proof of the existence

The excellent <https://physics.aps.org/articles/v10/5> and decent <https://en.wikipedia.org/wiki/Time_crystal#Experiments> (the former is [5] in that wikipedia article) are good starting points, with ample references.