Cosmological stasis and the coupled dark sector of the Dark Dimension

We study cosmological stasis in coupled dark matter and dark energy models in which a scalar field drives the dark energy and sets the dark matter mass, with the Dark Dimension as the motivating realization. Treating dark matter, dark energy, baryons, and radiation as an autonomous dynamical system, we show that exact stasis requires exponential slopes along the trajectory, and that running slopes give exact stasis or a drifting quasi-stasis depending on whether the running is integrable. The constant abundance solutions are the stationary solutions of coupled quintessence, which we read as stasis epochs with computable lifetimes and exits. Anchoring the cosmology to the measured abundances at matter-radiation equality and today, we show that the stasis region cannot accelerate for any initial conditions, and that the total equation of state is never phantom although the effective dark energy of a Λ CDM fit can appear so. For the Dark Dimension, capture into stasis in the future is excluded for geometric sources of the potential once dark fifth-force bounds are imposed, and dark matter production from the Standard Model brane supports a further stasis which the standard scenario does not reach. The stasis solutions provide analytic exclusions of the parameter space which do not depend on initial conditions.