Once upon a time some words meant something different than they do now. That’s true for a lot of words, “silly” once meant “blessed and worthy”. And “prove” once meant “test”.
But then mathematics came along and gave a whole new meaning to “prove”. A mathematical “proof” is absolute. It means there is no other way for things to be if your assumptions are true then the conclusion cannot be anything other than true. That’s an insanely high bar and can only ever apply in mathematics and logic. In a complex and physical “real” world such certainty is not possible, there are pretty much always other possibilities, even if they are pretty absurd. But sometimes the reality is absurd!
That mathematical meaning of ‘prove’ spilled over into our everyday speech however, to create a muddle, because we try to apply that meaning to the old meaning. Sometimes that’s harmless… the old saying “The proof of the pudding is in the eating” makes almost as much sense when you use the word in this extreme way rather than the original “test”. Sometimes it makes things kind of obscure but also interesting… another old saying “It’s the exception that proves the rule” gets some extra resonance when “proof” means something more than “test”. But sometimes it makes the meaning completely different and kind of broken. Like “has quantum theory been proven?”.
Now in the old sense of the world that question is very reasonable and the answer is absolutely yes! Quantum mechanics has been tested over and over, to extreme precision. It has met all of those tests beautifully. It is extremely well tested.
But in the new sense that question has no meaning. Theories are not mathematical theorems, they are explanations about why we see the things we do, and a means to predict what other things we’ll see under other circumstances. You can’t mathematically prove something like that any more than you can fry an adjective or conjugate peanut butter. There are always other possibilities that can explain the same observations, just as an infinite number of mathematical functions can fit a finite set of points!
For example there is an entirely separate theory, Bohmian Mechanics, which gives the same predictions as Quantum mechanics and yet which most people would consider a separate and different theory. I am sure there are infinitely many other theories that give the same predictions (within the parts of nature we can access with our experiments at this time).
(Above are possible electron trajectories in the double slit experiment according to Bohmian mechanics. Image from Wikipedia Commons)
The thing is that, because Quantum mechanics is so very well tested, if we are going to take any of those theories seriously then they will have to pass those same tests. For example it has to match the pattern of measurements in a double slit experiment… and all the tens of thousands of other experiments that have been done. Which means all alternative theories will be almost exactly the same as Quantum mechanics in most respects, including the weirdness that some people don’t like.
Me, I think weird is wonderful, but I’m a bit weird myself.
A theory is never proven, just tested and successful or unsuccessful at explaining what is seen. But Quantum mechanics is so well tested that anything that replaces it will look almost exactly the same in most ways.