times larger than antideuterons at 1 GeV/n, while the opposite is true of the annihilation cross-sections. In principle this implies a proportionally larger tertiary contribution for antihelium, where nuclear excitations remove kinetic energy during scattering. In order to determine the relevance of this, one must also estimate the typical number of scatterings during propagation. Assuming a cosmic-ray residence time $\tau_{\mathrm{res}} \approx 5\times10^6 ~\mathrm{yr}$ [@res_time](which is only a weak function of rigidity, scaling at most as $\mathcal{R}^{-0.6}$) [@res_time], a mean hydrogen density $n_H=1$ cm$^{-2}$, and a typical interaction cross-section $\sigma\approx 100$ mb, the number of scatters can be found by comparing the residence path length $c~\tau_{\mathrm{res}}$ with the mean free path $\lambda$:

$$\begin{aligned}
N_{\mathrm{scatters}} = \frac{c \tau_{\mathrm{res}}}{\lambda}=c \tau_{\mathrm{res}}~n_H \sigma\approx 0.2.\end{aligned}$$

With only a 20% chance of scattering, and given the small amount of energy removed during the inelastic process, we ignore all tertiary contributions in our semi-analytic treatment of interstellar propagation.

To bracket the impact of uncertainty in the anti-nucleus – proton cross-section, we use two methods: MethodANN and MethodINN which use the annihilation and total-inelastic cross-sections respectively in Eq. (\[eq:gamma\_int\]). For [$^3\overline{\text{He}}~$]{}, MethodINN leads to roughly a 40% lower flux than MethodANN, while for [$\overline{\text{D}}~$]{}, the results are nearly indistinguishable 