The spline fit of Φ µ (3) , µ (4) resulting from sampling ≈125,000 GBOMs can be found in Fig. , whereas the raw data prior to fitting the spline can be found in SI Sec. VI. Close to the Gaussian limit (µ (3) = 0, µ (4) = 0), the third order contribution should only provide a small correction to the second order cumulant response function, as all higher order cumulants strictly vanish for Gaussian fluctuations. As such, we expect that the largest prefactor values of Φ µ (3) , µ (4) ≈ 1 should reside close to the Gaussian limit, as is indeed observed. In this region, the undampened third order cumulant approximation provides the best approximation to the exact spectrum.