Prompt cusps in hierarchical dark matter halos: Implications for annihilation boost

Recent simulations have identified long-lived ``prompt cusps'' -- compact remnants of early density peaks with inner profiles $\rho\propto r^{-3/2}$. They can survive hierarchical assembly and potentially enhance signals of dark matter annihilation. In this work, we incorporate prompt cusps into the semi-analytic substructure framework SASHIMI, enabling a fully hierarchical, environment-dependent calculation of the annihilation luminosity that consistently tracks subhalos, sub-subhalos, and tidal stripping. We assign prompt cusps to first-generation microhalos and propagate their survival through the merger history, including an explicit treatment of cusps associated with stripped substructure. We find that the substructure hierarchy converges rapidly once a few levels are included, and that prompt cusps can raise the total annihilation boost of Milky-Way--size hosts at $z=0$ to $B\sim 50$ for fiducial cusp-occupation assumptions, compared to a subhalo-only baseline of $B_{\rm sh}\sim\mathrm{few}$. Across a wide range of host masses and redshifts, prompt cusps increase the normalization of $B(M_{\rm host},z)$ while largely preserving its mass and redshift trends. Compared to universal-average, peak-based estimates, our fiducial boosts are lower by about a factor of a few, primarily reflecting a correspondingly smaller inferred cusp abundance in host halos, highlighting the importance of unifying peak-based cusp formation with merger-tree evolution and environmental dependence.