  In this series of papers, we present a set of methods to revive quantum
geometrodynamics which encountered numerous mathematical and conceptual
challenges in its original form promoted by Wheeler and De Witt. In this paper,
we introduce the regularization scheme on which we base the subsequent
quantization and continuum limit of the theory. Specifically, we employ the set
of piecewise constant fields as the phase space of classical geometrodynamics,
resulting in a theory with finitely many degrees of freedom of the spatial
metric field. As this representation effectively corresponds to a lattice
theory, we can utilize well-known techniques to depict the constraints and
their algebra on the lattice. We are able to compute the lattice corrections to
the constraint algebra. This model can now be quantized using the usual methods
of finite--dimensional quantum mechanics, as we demonstrate in the following
paper. The application of the continuum limit is the subject of a future
publication.
