  The celebrated geodesic congruence equation of Raychaudhuri, together with
the resulting singularity theorems of Penrose and Hawking that it enabled,
yield a highly general set of conditions under which a spacetime (or, more
generically, a pseudo-Riemannian manifold) is expected to become geodesically
incomplete. It is a non-trivial question to ask whether (and to what extent)
there exist equivalently general conditions under which one expects discrete
spacetimes to become geodesically incomplete, and how these conditions might
differ from those in the continuum. This article builds upon previous work, in
which the conformal and covariant Z4 (CCZ4) formulation of the Cauchy problem
for the Einstein field equations, with constraint-violation damping, was
defined in terms of Wolfram model evolution over discrete (spatial) hypergraphs
for the case of vacuum spacetimes, and proceeds to consider a minimal extension
to the non-vacuum case by introducing a massive scalar field distribution,
defined in either spherical or axial symmetry. Under appropriate assumptions,
this scalar field distribution admits a physical interpretation as a collapsing
(and, in the axially-symmetric case, uniformly rotating) dust, and we are able
to show, through a combination of rigorous mathematical analysis and explicit
numerical simulation, that the resulting discrete spacetimes converge
asymptotically to either non-rotating Schwarzschild black hole solutions or
maximally-rotating (extremal) Kerr black hole solutions, respectively. Although
the assumptions used in obtaining these preliminary results are very strong,
they nevertheless offer hope that a more general, perhaps ultimately
``Penrose-like'', singularity theorem may be provable in the discrete spacetime
case too.
