The hypothesis that turbulence evolution can be described by kinetic energy, dissipation, and the Reynolds stresses alone is the basis of current second moment closures. This hypothesis is shown to be inconsistent with the Navier–Stokes equations at the two-component limit, because it implies that initially two-component turbulence remains two-component for all times. The inconsistency can be traced to the Markovian stress evolution imposed by the hypothesis; even Markovian spectral closures cannot predict evolution from a two-component initial state. Some implications of these findings for imposing realizability constraints on turbulence models are discussed.