Non-Causal Link- the belief in something unprovable

I was just thinking, is it actually possible to prove that two things do not share a casual link? Classically this is called "coincidence," but technically coincidence means occurring at the same time; I'm specifically referring to non-causal coincidence, where two things happen in the same timeframe but are not connected.

I would imagine this is as difficult as proving any negative with such an unknown field of variables. Two things happening at the same time might have an unseen connection, and the only way to actually prove that they do not have any unseen connections would be to trace every possible connection associated with each thing and confirm that there are no links. It seems like being certain, having proof, that two coincidental events are not connected would actually require omniscience, but in a state of omniscience no proof would be necessary anyway. Unless this state of non-causality can be proven, then claiming that two things that happened at the same time are not connected is actually a matter of belief. This would mean that one believing that the two things are connected, but has not proven this to be the case, and the one believing that two things are not connected, each believe that their stance on the matter is correct. The only difference is that one can prove that two things are linked with the resources that are available to mankind, but one cannot reasonably prove that two things are not linked (proving the negative in this example). So, in terms of taking a stance that can potentially be proven correct, it is actually the one that believes that the two things are linked who is closer to the state of being proven correct than one who believes that the two things are not linked.

I understand that in many cases the most popular opinion is to believe that two simultaneous things that have no obvious connection are non-causally coincidental, and that those that carry these beliefs often claim their belief to be fact. I also understand that group dynamics often result in peer pressure, and at times even bullying people, into accepting this belief- of non-causal relationships between things that are difficult to be explained as connected- as if it were proven fact. Without omniscience, claiming this certainty, this proof of a negative when the variables are not entirely known, is false. I have also seen cases where such a rigid/surface view of the world does not share a domain with a belief in God. Without the belief in God, presumably the possiblity of a position of omniscience from a person's internal belief system is also removed, so interestingly this combination of beliefs is in and of itself paradoxical (believing in the specific proof of a negative in so far as believing two things are non-causal coincidence, while simultaneously not believing that a state of omniscience- which is the only conceivable state where proof of this negative, given its myriad of unseen potential connections, could actually be known- is possible). 

I do not imagine I will sway any hearts or win any minds with this logical examination, in fact I imagine those holding such beliefs may become incensed by the logic presented here, perhaps incorrectly claiming it to be a logical fallacy, a philosophical trick. The truth is though that this is a fundamentally undeniable wedge in a logical system considered to be entirely and perhaps imperviously logical, however small of a wedge it might be. Still, raising the shield of emotion does not prove a thing to be false, not even within the confines of one's own mental structure. Claiming absurdity does not stop a seed from being planted. One's beliefs, regardless of how consistently reinforced or broadly opposed these beliefs are, does not make the thing believed to be fact or not- presumably the thing is fact or not fact by its own merit, independent of belief. Such beliefs remain belief, until such time that the truth of something becomes actually known (seemingly a sticky issue in and of itself), or until the belief is set aside, or burned away.