Our intent in this appendix is to provide a linkage to the two previous papers the authors have written on restricted orbit equivalence. We say that an r-size is a size function m as defined in this current work, in Section 2.2. (The “r” denotes “rearrangement”.) We will see that the notion of m-equivalence developed in (where m is a p-size, where “p” denotes “permutation”) is subsumed by our work here. On the other hand, we will not quite be able to show this for the work in. What we will see though is that a slight strengthening in the definition of the equivalence relation associated with a 1-size, (size as defined in) will make it possible to describe the equivalence relation as a restricted orbit equivalence in the sense we describe here. As we will see, this change will have no effect on the examples described in, and the m-f.d. systems for the original equivalences are unchanged by this strengthening. Whether some equivalence classes of arrangements are possibly changed for some 1-size, we do not know. As we now consider the definition in to have been a very preliminary, perhaps unrefined, attempt to axiomatize the notion of restricted orbit equivalence, we have not pursued this issue further. Our main interest here is to bring the examples, in particular examples 3 and 4 (referred to as mφ and mφ) under the umbrella of our work here.