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Explaining the complex structure and dynamics of sleep, which consist of alternating and physiologically distinct nonREM and REM sleep episodes, has posed a significant challenge. In this study, we demonstrate that a single wave model concept captures the distinctly different overnight dynamics of the four primary sleep measures—the duration and intensity of nonREM and REM sleep episodes—with high quantitative precision for both regular and extended sleep. The model also accurately predicts how these polysomnographic measures respond to sleep deprivation or abundance. Furthermore, the model passes the ultimate test, as its prediction leads to a novel experimental finding—an invariant relationship between the duration of nonREM episodes and the intensity of REM episodes, the product of which remains constant over consecutive sleep cycles. These results suggest a functional unity between nonREM and REM sleep, establishing a comprehensive and quantitative framework for understanding normal sleep and sleep disorders.