  A while ago a proposal have been made regarding Klein Gordon and Maxwell
Lagrangians for causal set theory. These Lagrangian densities are based on the
statistical analysis of the behavior of field on a sample of points taken
throughout some "small" region of spacetime. However, in order for that sample
to be statistically reliable, a lower bound on the size of that region needs to
be imposed. This results in "unwanted contributions" from higher order
derivatives to the Lagrangian density, as well as non-trivial curvature effects
on the latter. It turns out that both gravitational and non-gravitational
effects end up being highly non-linear. In the previous papers we were focused
on leading order terms, which allowed us to neglect these nonlinearities. We
would now like to go to the next order and investigate them. In the current
paper we will exclusively focus on the effects of higher order derivatives in
the flat-space toy model. The gravitational effects will be studied in another
paper which is currently in preparation. Both papers are restricted to bosonic
fields, although the issue probably generalizes to fermions once Grassmann
numbers are dealt with in appropriate manner.
