 Freiman lifting of a progression

An eight-term Freiman isomorphism extends injectively to the two-sum/two-difference set by sending its representations to the corresponding integer differences. Independence of representation uses four-term relations, and additivity whenever $y,z,y+z$ lie in that difference set uses six-term relations. Thus a proper symmetric progression in the modeled difference set lifts, one coordinate step at a time, to a proper integer progression of the same rank and cardinality.