The GBOM is a convenient, numerically robust model in which the second order cumulant approximation is no longer exact, but in which the exact second and third order cumulants can be evaluated analytically (see SI Sec. II and Refs. 39 and 60). Furthermore, the exact response function (Eqn. 5) can be computed analytically, enabling us to examine in detail the errors introduced by truncating the cumulant expansion at some finite order.