The spatially-independent spectrum of neutrons slowing down in an infinite medium with constant cross sections is calculated from both the Laplace transform exactitude (LTE) and a generalized synethtic kernel approximation (SKA). The fluxes are expressed as sums of exponentials in lethargy and compared asymptotically. For hydrogenous mixtures, two of an infinite number of terms from the LTE are non-oscillatory, both dominate all others at large lethargies, and one vanishes whenever hydrogen is a sole or missing constituent. The SKA yields a solution consisting of as many exponentials as isotopes present. The longest-lived terms are generally most accurate, but even the dominant one can be exact only if there is no absorption or if hydrogen is the sole moderator. For binary mixtures, both terms in the SKA fluxes are non-oscillatory, and the secondary one vanishes for the same concentrations that make the corresponding term in the LTE vanish. Analytic expressions for errors in the asymptotic flux from the SKAs are given as a function of lethargy, all the cross sections, and masses. For every instance observed, the exact asymptotic flux is bounded on different sides by values from the Greuling-Goertzel and Selengut-Goertzel SKAs.