  In part III is realized the consistent development of Heisenberg--Dyson's
two-layer matrix approximation to the graph formalism for postulating discrete
physical objects (DPO) introduced in parts I-II in the form of discrete sets of
graphs--skeleton ({\bf SvT}) or root ({\bf RvT}) $v$-trees, beyond common
space--time. It is noted that already in the late-1950s one made an attempt to
formulate in physical theory the discontinuity as an element of some special
diagram technique. In the framework of pointed Heisenberg--Dyson's two-layer
matrix scheme, with an incidence {\bf I} and a loop {\bf CD}$(\delta)$ graph
matrices, are got the following main results: (1) the many-``planes'' {\bf SvT}
or {\bf RvT} representation of any DPO in opposition to one-``plane'' physical
objects in continuous physical models; (2) the superposition of different types
of interaction for any microobject where {\bf RvT} representation for
short-ranged interactions (weak, strong) is one-``plane'' and for long-ranged
interactions (gravitational, electromagnetic) is many-``planes''; (3) based on
the incidence matrix {\bf I} (upper layer) ``graph geometry'' of real DPO
describes their peculiar many-``planes'' inner structure beyond common
space--time; (4) the notion of interacting ``charge'' can be extracted only
from the symbolical quantities for the quasi-continuous field ``objects'' by
means of the loop matrix {\bf CD}($\alpha$) (under layer); and some other
concrete results of an analysis of structural peculiarities of DPO.
