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A dispersive decay estimate is proven for the linear propagator of the three-scale rotating stratified Boussinesq equations in the stratification-dominant regime. Because the phase function of that propagator is both singular and degenerate, obtaining the estimate requires novel techniques involving cutting phase-space shells into several pieces and using different methods in each region. Using the decay estimate, we prove the long-time existence of solutions to the initial-value problem for the nonlinear equations and obtain a rate of convergence to the limit system and intermediate asymptotics. [ABSTRACT FROM AUTHOR]