Breaking Neutral Theory: Empirical Falsification of Effective Population-Size Invariance in Kimura's Fixation Model

Kimura’s neutral theory includes the famous invariance result: the expected rate of neutral substitution equals the mutation rate μ, independent of population size. This result is presented in textbooks as a general discovery about evolution and is routinely applied to species with dramatically varying population histories. It is not generally true. The standard derivation holds exactly only for a stationary Wright-Fisher population with constant effective population size. When population size varies—as it does in virtually every real species—the expected neutral substitution rate depends on the full demographic trajectory and is not equal to μ. We demonstrate this mathematically by showing that the standard derivation uses a single symbol (N_e) for two distinct quantities that are equal only under constant population size. We then show that the direction of the predicted deviation matches observed patterns in three independent mammalian comparisons: forest versus savanna elephants, mouse versus rat, and human versus chimpanzee. Kimura’s invariance is an approximation valid only under demographic stationarity, not a general law. Evolutionary calculations that apply it to species with changing population sizes are unreliable.