What is the relationship of the versions of the male-female saying of Jesus?

Richard has brought up in his comment to my last Apocryphote on the Gospel of the Egyptians the question what was the relationship of the versions to each other? The problem has traditionally been approached from source, form, and redaction critical methods. And it has essentially got us no closer to a good answer.

My approach to the problem is broader and based on the movement of traditions around early Christian communities. First, some version of this saying was an oral teaching that became a liturgical prayer at baptism long before it was written down. So we don't have any "originals" in terms of the originating version, but we have four "originals" in terms of the moment that they were written down. None of them appear to be literary copies of the other. So there is no literary dependence going on. In fact, what we see is a very "oral" consciousness here, with bits of different sayings that have twined together on different occasions and for different reasons.

Compare for instance saying 37 of the Gospel of Thomas which I have put up as the Apocryphote today. See how pieces of that saying are also in the Gospel of Egyptians version of the male-female saying? This is what happens when materials pass around in oral environments or environments where human memory is the operating mechanism (not literary copying). Pieces of sayings get confused because they become attached to other sayings in people's memories. So when they are recalled from memory, particularly orally where there is no opportunity to correct or revise, they become mixed up.

So what can we know about the transmission history of the male-female saying? Gal. 3 is our oldest version. It is liturgical, which means it is being used as part of the baptism ceremonies in Antioch and probably has been for a long time prior to Paul's arrival. The saying pops up in the accretions of the Gospel of Thomas in Syria in the late first century probably because it was being used in liturgy there too. The saying is known also in Alexandria to both Clement of Alexandria in the late second century and in the Gospel of the Egyptians from the early or mid-second century. And the saying is known to 2 Clement, a mid-second century Roman (?) homily written on for a baptismal service (?).

Tracking this as a tradition, what is the common denominator? It is an early liturgy that Paul knows was being used in Antioch where he learned it. My best critical guess is that this saying and the liturgy upon which it was based was from Jerusalem which had a mission to Antioch, Edessa (Gospel of Thomas), Alexandria (Gospel of the Egyptians), and Rome (2 Clement). So the saying and liturgy were well-distributed from an early time. I want to also note that Jerusalem sits on the road between Edessa and Alexandria. So there is a major connection in the early Christian traditions between Syrian and Alexandrian Christianity which develop quite independent of Rome for a couple of centuries.