  This article describes the regularization of the generally relativistic gauge
field representation of gravity on a piecewise linear lattice. It is a part of
the program concerning the classical relativistic theory of fundamental
interactions, represented by minimally coupled gauge vector field densities and
half-densities. The correspondence between the local Darboux coordinates on
phase space and the local structure of the links of the lattice, embedded in
the spatial manifold, is demonstrated. Thus, the canonical coordinates are
replaceable by links-related quantities. This idea and the significant part of
formalism are directly based on the model of canonical loop quantum gravity
(CLQG).
  The first stage of this program is formulated regarding the gauge field,
which dynamics is independent of other fundamental fields, but contributes to
their dynamics. This gauge field, which determines systems equivalence in the
actions defining all fundamental interactions, represents Einsteinian gravity.
The related links-defined quantities depend on holonomies of gravitational
connections and fluxes of densitized dreibeins. This article demonstrates how
to determine these quantities, which lead to a nonpertubative formalism that
preserves the general postulate of relativity. From this perspective, the
formalism presented in this article is analogous to the Ashtekar-Barbero-Holst
formulation on which CLQG is based. However, in this project, it is
additionally required that the fields' coordinates are quantizable in the
standard canonical procedure for a gauge theory and that any approximation in
the construction of the model is at least as precisely demonstrated as the
gauge invariance. These requirements lead to new relations between holonomies
and connections, and the representation of the densitized deibein determinant
that is more precise than the volume representation in CLQG.
