“…We demonstrate theoretically and through examples that mathematical constraints on transforming and averaging taxon trajectories through time have important implications for what can and cannot be derived from these trajectories in terms of mechanisms and patterns of changes in range size (Willis, 1922; Gaston, 1998; Webb & Gaston, 2000), occupancy, and diversity through time (Gould, Gilinsky & German, 1987; Foote, 2007; Foote et al, 2007; Liow & Stenseth, 2007; Pigot, Owens & Orme, 2012; Plotnick & Wagner, 2018). The constraints apply to all research questions in which symmetric waxing and waning has emerged and served to infer on macroecological patterns.…”