Our knowledge of reality accumulates in fits and starts, sometimes slowly progressing as we fill in details about things we're pretty sure exist, sometimes stumbling upon the entirely unexpected, and sometimes going out hunting for snarks.
A snark is a theoretical structure, effect, or object that we infer exists, but haven't actually found.
Radio waves were once a snark: they were predicted by Maxwell's electromagnetic equations, but never seen until Hertz discovered them.
In other cases, our snarks are more speculative. Possibly boojum.
As physicists started to apply sophisticated mathematical concepts from group theory to weak and electromagnetic interactions in the 1950s they recognized they had a problem: there were some ideas that seemed to work really well... but only if none of the particles involved had mass. Since things like electrons and protons and neutrons manifestly do have mass, this suggested the whole theoretical approach was a dead end, until in the early 1960s Peter Higgs and others realized that if there was a special kind of field that permeated the whole universe and coupled with particles that experienced the weak nuclear force, it could give them mass without breaking the rest of the theory.
Fifty years later the Higgs boson was discovered at CERN, the large European particle accelerator lab, and what had been a wholly theoretical entity--what I'm calling a snark--became an piece of ordinary physics. Today, new experiments are aimed at measuring the parameters of the Higgs more precisely, now that we know it actually exists.
Snarks are exciting because they are make-or-break moments for theories: if a theory predicts the existence of something, and it does not in fact exist, the theory gets knocked on the head. It may yet get up from the mat and get back into the fight, but coming back from a blow like that is hard.
We never prove that something exists or does not exist: we produce evidence of existence or non-existence. Evidence of existence is easy: effects are evidence of causes. When physicists at CERN saw an excess of scattering events with the right signature in the mass range where they expected to find the Higgs, they had an effect that the Higgs was the most plausible cause of.
Evidence of non-existence is harder, but still straightforward: you have to show that if a thing existed, it would have some effect, and when the effect fails to show up in an apparatus that is demonstrably sensitive to it, that is evidence for the non-existence of that thing.
But in some cases we can't just go looking for such effects. The circumstances where they would come into play are too extreme, or too unlikely, or simply too far away, for us to be able to look directly. In those cases we can, sometimes, come up with an analogous system that follows the same mathematical description, and use it to test our thinking.
The thinking that goes into our mathematical descriptions of theoretical entities is subtle and complex and inherently non-mathematical: physics is not math. Math is a tool that physicists use to think about reality, but it's one of many. It's a good tool because the formal properties of mathematics make it relatively easy to check, but if the foundational, physical, thinking is wrong we may wind up with math that we think describes a black hole or a magnetic monopole or one of the menagerie of particles implied by string theory, but which actually does nothing of the kind.
One of the most challenging and troubling theoretical entities in modern physics is Hawking radiation, which is a quantum-mechanical phenomenon that is visible around black holes and other high-acceleration environments. The idea is that the what we think of as "empty" space is in fact a sea of short-lived virtual particles that continuously appear and disappear. We can measure the effects of this sea directly in the form of the Casimir force between conducting plates that are extremely close to each other.
When a pair of virtual particles appears near the event horizon of a black hole, it is possible for one member of the pair to appear below the horizon and the other above it. But it's a property of the horizon that nothing below it can ever escape, which means that particle falls in, so the other particle is left, unbalanced, in our universe, and is emitted from the black hole.
There are all kinds of problems with this idea, which eventually led to the "firewall" theory of black holes that posits there must be a region just within the event horizon that effectively obliterates any infalling matter.
There are other problems as well, none of which can be studied directly because all the black holes we know about are--thankfully--too far away. But it turns out we can recreate the physics of a black hole in the laboratory using an ultra-cold gas, one part of which is stationary, adjacent to another part that is flowing faster than the speed of sound in the gas.
When this happens, pairs of sound quanta--called "phonons"--can appear spontaneously at the boundary between the two regions. These are the acoustic equivalent of virtual particles in empty space. They always appear in pairs because it's the only way conservation laws can be respected. The boundary acts like the event horizon of a black hole: one member of the pair gets swept away by the super-sonic flow, while the other can radiate off, away from the boundary.
By detailed study, we can confirm that important features of Hawking radiation do in fact occur in this analogue system, and that makes the problems encountered by the theory even more pointed... but hopefully some fancier analogue will allow us to investigate those problems in the lab, and point us the way to a solution.
Unless the snark really is boojum, and our models are missing some essential element of reality that only more observation of the real system will reveal.