  The physical meaning, the properties and the consequences of a discrete
scalar field are discussed; limits for the validity of a mathematical
description of fundamental physics in terms of continuous fields are a natural
outcome of discrete fields with discrete interactions. The discrete scalar
field is ultimately the gravitational field of general relativity, necessarily,
and there is no place for any other fundamental scalar field, in this context.
Part of the paper comprehends a more generic discussion about the nature, if
continuous or discrete, of fundamental interactions. There is a critical point
defined by the equivalence between the two descriptions. Discrepancies between
them can be observed far away from this point as a continuous-interaction is
always stronger below it and weaker above it than a discrete one. It is
possible that some discrete-field manifestations have already been observed in
the flat rotation curves of galaxies and in the apparent anomalous acceleration
of the Pioneer spacecrafts. The existence of a critical point is equivalent to
the introduction of an effective-acceleration scale which may put Milgrom's
MOND on a more solid physical basis. Contact is also made, on passing, with
inflation in cosmological theories and with Tsallis generalized one-parameter
statistics which is regarded as proper for discrete-interaction systems. The
validity of Botzmann statistics is then reduced to idealized asymptotic states
which, rigorously, are reachable only after an infinite number of internal
interactions . Tsallis parameter is then a measure of how close a system is
from its idealized asymptotic state.
