  Hamilton's Ricci flow (RF) equations were recently expressed in terms of a
sparsely-coupled system of autonomous first-order nonlinear differential
equations for the edge lengths of a d-dimensional piecewise linear (PL)
simplicial geometry. More recently, this system of discrete Ricci flow (DRF)
equations was further simplified by explicitly constructing the Forman-Ricci
tensor associated to each edge, thereby diagonalizing the first-order
differential operator and avoiding the need to invert large sparse matrices at
each time step. We recently showed analytically and numerically that these
equations converge for axisymmetric 3-geometries to the corresponding continuum
RF equations. We demonstrate here that these DRF equations yield an explicit
numerical realization of Thurston's geometrization procedure for a discrete 3D
axially-symmetric neckpinch geometry by using surgery to explicitly integrate
through its Type-1 neck pinch singularity. A cubic-spline-based adaptive mesh
was required to complete the evolution. Our numerically efficient simulations
yield the expected Thurston decomposition of the sufficiently pinched axially
symmetric geometry into its unique geometric structure -- a direct product of
two lobes, each collapsing toward a 3-sphere geometry. The structure of our
curvature may be used to better inform one of the vertex and edge weighting
factors that appear in the Forman's expression of Ricci curvature on graphs.
