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Neanderthals were rapidly made extinct, along with archaic Homo sapiens, by the invasion of new Homo sapiens, sometimes referred to as Cro Magnon, who were the first "Humans" having power of cognitive thought (i.e., abstract thought including language). This was accompanied by a critical genetic marker known as the D-allele.
As we know, there was a limited amount of cross-breeding; but, it is more likely that was between Neanderthal and archaic Homo sapiens, who then had limited reproductive contact with the new, genius breed.
This invasion of our mentally-modern (as opposed to anatomically modern) human ancestors entered the world in north-central Eurasia somewhere around 40,000 years ago.
human ancestors entered the world in north-central Eurasia somewhere around 40,000 years ago.
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And had been around for some 260,000 years prior, according to mitochondrial DNA.