I want less leakage. 22nH or less leakage would be nice.
I figure in order to reduce the leakage I need to get my secondary and primary closer together. iows I need coax with less inner insulator or perhaps switch to parallel bonded magnet wire or go twisted pair. True? Or maybe I'm overlooking something.
Actually, you probably are. Try a non-progressive winding - count the number of turns, then go through the four cores putting on half that number of turns before coming back through the four cores putting on the rest of the turns in the same sense, so that the end of the coaxial cable emerges from same point as where the widing started.
At the moment you have a - very weak - external field produced by the imperfectly coaxial nature of your coaxial cable as it makes its way around the loop in the plane defined by the fours cores; with the non- progressive winding this loop is much reduced.
The "imperfection" does seem to be about 0.1% - a non-progressive winding might well give you your next factor of ten reduction.
Progressive versus non-progressive windings are discussed in Rayner and Kibble's "Coaxial AC Bridges" a.k.a."Tricks from the National Standards Labs".
I've been thinking about it, and you might be able to do some good with less work.
Setting the cores on edge, rather than laying them flat - as pictured
- would reduce the are included in the loop that presumably is responsible for your leakage inductance.
Twisting that loop could make the effective area even smaller.
If we label the four cores as A,B,C and D and the present arrangement is
A B you might do better with A B D C C D.
Worth a try.
Granting that this wouldn't be problem if coaxial cable was perfectly symmetrical, one could argue that the imperfections in symmetry that are messing you about are randomly distributed, and thus the re- arrangement can't be assumed to cancel the stray coupling.
This may be true, but it is likely that bent coaxial cable isn't perfectly symmetrical in regular and reproducible way which would make the stray field more predictable and more likely to cancel.