In a big enough, old enough universe, random fluctuations should assemble lone brains with false memories far more often than whole ordered worlds. Statistically, you should be one.
Documented
The reductio that became a research program
Boltzmann proposed our region's order as a rare statistical fluctuation in an eternal high-entropy universe; the objection, sharpened a century later, is that the smallest fluctuation sufficient for an observer is a single brain with fabricated memories, so fluctuation cosmologies predict most observers are such brains, contradicting our ordered observations. Modern cosmology treats this as a live constraint: eternal inflation and certain de Sitter models must show their Boltzmann-brain production stays sub-dominant or be ruled out.
Assessment: Documented and serious: the counting argument is valid probability, and papers by Dyson, Kleban, Susskind, Carroll, and others genuinely use it to test models. It is the only paradox that functions as quality control for cosmology.
Stories
The void-brain as meme
The folk version flattens it into a creepypasta: you are a brain that popped into existence three seconds ago, complete with this sentence in your false memory. It circulates as a dorm-room nihilism tool, unfalsifiable and unbeatable in argument, which is exactly why it wins no converts and ruins many evenings.
Assessment: As folklore it is documented; as a claim about you it is self-undermining, since a Boltzmann brain's reasoning, including this paradox, is unreliable by its own premise. The position cannot be argued into, only worried into.
Interesting claims
The measure problem underneath
The interesting claim is the one cosmologists actually sweat: the paradox exposes that we have no justified way to count observers across an infinite universe, the measure problem. Solve the brains and you solve how to do probability in cosmology at all; fail, and every prediction including this one floats on convention.
Assessment: Documented as an open foundational problem. The brains are not the threat; the missing mathematics of typicality is. The paradox endures because it is the cleanest proof that infinity breaks our intuitions in ways we have not repaired.
Sources and further reading
Boltzmann's replies to Zermelo (1896-97), the origin
Dyson, Kleban, Susskind, Disturbing Implications of a Cosmological Constant (2002)
Carroll, Why Boltzmann Brains Are Bad (2017), the cognitive-instability argument
Albrecht and Sorbo, Can the Universe Afford Inflation? (2004), the measure constraint
Page, the de Sitter decay calculations, 2006 onward
Worth asking
You cannot prove you are not a Boltzmann brain without trusting memories a Boltzmann brain would also have. What exactly is your evidence doing for you?