  We present the first 3-dimensional, fully compressible gas-dynamics
simulations in $4\pi$ geometry of He-shell flash convection with proton-rich
fuel entrainment at the upper boundary. This work is motivated by the
insufficiently understood observed consequences of the H-ingestion flash in
post-AGB stars (Sakurai's object) and metal-poor AGB stars. Our investigation
is focused on the entrainment process at the top convection boundary and on the
subsequent advection of H-rich material into deeper layers, and we therefore
ignore the burning of the proton-rich fuel in this study. We find that, for our
deep convection zone, coherent convective motions of near global scale appear
to dominate the flow. At the top boundary convective shear flows are stable
against Kelvin-Helmholtz instabilities. However, such shear instabilities are
induced by the boundary-layer separation in large-scale, opposing flows. This
links the global nature of thick shell convection with the entrainment process.
We establish the quantitative dependence of the entrainment rate on grid
resolution. With our numerical technique simulations with $1024^3$ cells or
more are required to reach a numerical fidelity appropriate for this problem.
However, only the result from the $1536^3$ simulation provides a clear
indication that we approach convergence with regard to the entrainment rate.
Our results demonstrate that our method, which is described in detail, can
provide quantitative results related to entrainment and convective boundary
mixing in deep stellar interior environments with veryvstiff convective
boundaries. For the representative case we study in detail, we find an
entrainment rate of $4.38 \pm 1.48 \times 10^{-13}M_\odot \mathrm{/s}$.
