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Large language models show that language is learnable from data, but their scale and training conditions differ sharply from a child\u2019s natural experience. We trained cognitively feasible, child-specific models on longitudinal home recordings from eight children across their first 1000 days of life. Each model learned incrementally from a single child\u2019s ordered speech input without prior linguistic knowledge, developing phoneme-like and word-form representations that tracked each child\u2019s learning pace. Learning was dependent on dense daily sampling and replay of recent input, indicating that both experience- and memory-related processes are necessary. 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