We also note that there is a low skewness, high kurtosis region where the optimized third order cumulant method under-preforms the second order cumulant approximation. This behavior may be rationalized in the following way: Within this region, one observes both numerically unstable GBOM parameterizations and GBOMs for which the unaltered third order cumulant and second order cumulant are in close agreement. With respect to the first effect, we observe nonphysical artifacts in this region not only for the third order cumulant approximation, but also with FC and ensemble computational methods which are generally assumed to be stable. This suggests that this region likely corresponds to model parameterizations that lie outside the domain of realistic molecular systems. This observation is reinforced by the fact that none of the real molecular systems sampled with MD investigated in this study (see SI Sec. VII and Sec. IV C) reside within this domain of statistical space. In fact, with all molecules studied for this work we find that their energy gap fluctuations reside in regions where the optimized third order cumulant approximation is predicted to out- preform the second order cumulant approximation.