Could you prove something to be "onologically random"?
Probably not.
You may prove the inability to anticipate an event. I think that is Heisenberg.
Well let me make up a tree of randomness.
Randomness can be of two kinds: ontological - in terms of meaning - or methodological - in terms of practical predictability.
There are two kinds of methodological randomness: statistical randomness, and causal randomness. In both cases, we predict and manipulate the properties of an ensemble of observations, without knowing - or needing to know - the individual observations ahead of time. For example, on average, 1/6 rolls of a fair die will yield the number 1. On average, 10 people out of a thousand will die of car accidents each year in a particular country. On average, 1/2 of the radioactive nuclei in a particular sample will decay over one half-life.
But there's an important difference between something that is statistically random and something causally random. The first two examples are statistically random. We don't predict observations on an individual level not because we can't, but because we're too lazy to. For example take a die throw. In theory, I could look at the initial position of the die, the force applied by the thrower's hand, the air currents, the distance from the table, and the force of gravity, and plug all that into a huge computer which would then tell me precisely what number would come up. Or for car drivers: given a particular accident, I can tell you that the guy got into it because he was trying to shave in the rearview mirror when the traffic light in front of him suddenly turned red. Given his particular situation, I can predict that he'd suffer a bad crash, and if I had the same information on everything that every driver did every day before they got into the car I could tell you about their accidents or not as well.
Whereas for something causally random, there
is no previous trigger, no cause, no hidden variable. Something just ... happens. That's the case with radioactive atoms decaying, and excited electrons emitting light, and lots of other stuff. You can't tell which nucleus will decay, or when the electron will jump to a lower level, no matter how much information you collect about them.
Having said that, there is no reason to believe that a methodologically random event - something we don't, or can't predict beforehand - should also be an ontologically random event - something that has no meaning, purpose, or significance. Conception is random - try telling a parent that it's meaningless!
