Discrete fields and the Pioneer anomalous acceleration

The dominant contributions from a discrete gravitational interaction produce the standard potential as an effective continuous field. The sub-dominant contributions are, in a first approximation, linear on n, the accumulated number of (discrete) interaction events along the test-body trajectory. For a nearly radial trajectory n is proportional to the traversed distance and its effects may have been observed as the Pioneer anomalous constant radial acceleration, which cannot be observed on the nearly circular planetary orbits.