In fact, it would probably be impossible to prove that synchronicity really is an existing phenomenon, at least by using the normal methods of scientific deduction. This is because synchronicity operates completely outside the current laws of physics as defined by scientists. Jung was the one that created the word and also the theory on synchronicity. This would mean that if we had all the existing factors, we could calculate exactly what would happen in any given system. In the microverse, i.e. on a sub-atomic level, all the laws of physics are out the window.
Particles can be on two places at the same time.
Particles can travel back in time. Things happen for no reason, i.e. the effect has no cause. One cannot predict where a particle will go next, or even the behavior of an atom. In fact, also outside of the microverse it is now understood that a given cause does not always produce the same effect, although it might SEEM that way in most instances. Two events which are somehow connected, but neither of them caused the other one to happen.