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The Neolithic revolution initiated a pivotal change in human society, marking the shift from foraging to farming. The underlying mechanisms of agricultural expansion are debated, primarily between cultural diffusion (knowledge and practices transfer) and demic diffusion, or people migration and replacement. Ancient DNA analyses reveal significant ancestry changes during Europe's Neolithic transition, suggesting primarily demic expansion. However, the presence of 10-15% hunter-gatherer ancestry in modern Europeans indicates cultural transmission and non-assortative mating were additional contributing factors. We integrate mathematical models, agent-based simulations, and ancient DNA analysis to dissect and quantify the roles of cultural diffusion and assortative mating in farming's expansion. Our findings indicate limited cultural transmission and predominantly within-group mating. Additionally, we challenge the assumption that demic spread always leads to ancestry turnover. This underscores the need to reassess prehistoric cultural expansions and offers new insights into early agricultural society through the integration of ancient DNA with archaeological models.