Higher-order Compton scattering smoothing conjecture

Let $\operatorname{Spec}(E,\mathbf{d},\mathbf{s})$ denote the measured Compton-scattering spectrum, decomposed into contributions $g_i(E,\mathbf{d},\mathbf{s})$ from scattering of order $i$, and let $\mathcal{L}_i(\cdot,n_e)$ be the Fourier integral operator approximating $g_i$, where $n_e$ is the electron density. Thus

$$\operatorname{Spec}(E,\mathbf{d},\mathbf{s})=\sum_{i=1}^{\infty}g_i(E,\mathbf{d},\mathbf{s}).$$

Higher-order Compton scattering smoothing conjecture. For each scattering order $i$, $g_i$ is approximated by a Fourier integral operator $\mathcal{L}_i(\cdot,n_e)$ of order

$$-\frac{3i+1}{4}.$$

Consequently, the contribution to the spectrum becomes smoother as the number of scattering events increases. This is an intuitive extension of the analysis for the first and second scattering orders to the full spectrum; the geometry of higher-order scattering is harder to model or implement, so the statement is presented as a conjectural principle.