@Vern Edwards Just read your post on asking questions and I see that I had a few assumptions in this question.
I was brought up as modification authorities are either a) the clause from your contract that authorized the change; or if there was no clause than the only other option was b) Mutual agreement; as shown in this quote from the "DAU modification authority decision help guide":
"The vast majority of contract changes are effected by the authority of the existing terms and conditions (i.e., clauses) within a contract. When so, that particular clause shall be used as the authority for the change. In the rare event that a change is necessitated by a condition not specifically covered within a contract clause, then common practice is to state as the authority: “Mutual agreement of the contracting parties”. The checkbox in front of Block 13C would be checked, and this phrase would be typed in block 13C."
Further, as a minor point, it was just hammered into me and is now habit to use the part 43 cite when making mutual agreements; although I do agree that the cite is not required.
I see from the conversation that using 41 U.S.C. 4106(c)(2) and FAR 16.505(b)(2)(i)(B) isn't wrong to cite per se, although in DoD I feel that it would be "wrong" in that there would be few reviewers who would sign off on it; at least in the offices I worked. I'm not in those offices anymore though, so need to adjust my thinking.
Again, super interesting conversation here. Food for my thoughts.