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This paper demonstrates the feasibility of the hypothesis that Neandertals formed when a population using recently developed Levallois stone tool technology expanded between 400-250 thousand years ago (ka). In Europe, their range expansion into an area with Sima de los Huesos-like people led to massive introgression of local archaic genes producing a population with around 95% archaic ancestry (Neandertals); if this range expansion was sex-biased it would provide a simple explanation for why Neandertals retain modern human lineage Y chromosomes or mitochondrial DNA. In Africa, interbreeding with local archaic humans led to more modest archaic admixture and the deep substructure detected in all modern humans today. This proposal explains four previously perplexing similarities of modern humans and Neandertals-sharing of mitochondrial DNA, Y chromosomes, Levallois tools, and 300-200 ka date of formation by mixture-even while Neandertals and Denisovans cluster genome-wide.