The Unprovable Negative Fallacy is a fake logic rule stating that "It's impossible to prove a negative." or, alternatively "It's impossible to prove the non-existence of something.".
It's a completely made-up rule that's not found nor does derive from any system of formal logic, and in fact goes directly against some of the most basic and universally recognized laws of inference: specifically, by the law of double negation, *any* statement is the negative of its complementary statement; therefore, claiming that "negatives cannot be proved" is equivalent to claiming that *nothing* can be proved, rendering all of logic completely useless. For this reason in logic positives and negatives are by default perfectly equivalent, and in no way the latter deserves a "privileged" treatment compared to the former.
It also goes against centuries of progress in both mathematics and natural sciences, both of which make extensive use of negative proofs in countless contexts: stuff like the non-existence of a 6th regular polihedron or of a stable chemical compound with formula S2H, the impossibility of squaring the cirle or achieving a perpetual-motion machine, are just some examples of a potentially endless list of negative statements that are both clearly provable and rigorously proven.
In order to give the slogan a semblance of validity, the pseudo-rationalist often ironically asks to prove the non-existence of blatantly imaginary stuff like "unicorns", "Santa-Claus" or "a teapot orbiting around the Sun", with the tacit implication that no one can do such a thing; aside from the intrinsic dishonesty of comparing a legit belief with actual basis (however weak they might be) with something that is deliberately made-up on the spot and no one believes into, proving the non-existence of such things is actually not only possible, but also extremely easy for anyone, by just drawing from universally accepted scientific knowledge or reasoning about elementary probability.