I didn't mean to post a question and then answer it myself, but in my searches on Chabad.org I found this question and answer page which discusses why there is a custom to leave space between the third and fourth wraps of the tefillin shel yad.
Why is it Chabad custom to leave space between the third and fourth tefillin "wrap" around the arm?
The first three coils correspond to the three letters of the divine name ש-ד-י (Shaddai). The latter four coils correspond to the four letters of the divine name י-ה-ו-ה (the Tetragrammaton). Because these two groups of coils represent two different names, it is appropriate to leave a little space between them. The amount of space you should leave is not specified anywhere—just leave enough space so that the break will be noticeable.
Rabbi Eliezer Posner
So to summarize, the first three wraps apparently represent the divine Name of Shaddai and the last four wraps represent the divine Name of Hashem, the Yud-Kay-Vav-Kay.
I would still be interested in more answers, by the way.