You might want to consider theory around a fly-back transformer. When a core becomes saturated the magnetic field is distorted around the gap in a ferromagnetic material. This distortion will collapse yielding an additional energy. You will find a flyback transformer on CRTs. I just removed one last night to experiment with. An extra winding is put around the secondary to feed back this energy into the primary. This will boost the voltage output over a normally wound transformer. I wanted the ferromagnetic material more then I wanted the winding and circuits. I went down this rabbit hole because i am working on a magnetic loop antenna and wanted to know more about magnetic circuits. and ran across fly back transformer that used this property. It is a very real phenomena.
Oh No Mr Bill!, here we go again!, folks that don't appreciate the mysteries of the universe
might want to skip over this post!
the energy in a magnetic field per unit volume is
E = 1/2 B^2 / Mu,
B is the magnetic field (flux per unit cross sectional area),
Mu is the permeability of the medium
the permeability of iron is roughly 3 x 10^-3 Henries (times length divided by area)
the permeability of air is roughly 1 x 10^-6 Henries (times length divided by area)
so the energy in an air gap is larger per unit volume than in iron,
but the total energy depend on the total volume of each (intuitively, it takes more
energy to create a magnetic field in air than in iron, so there is more energy stored
in air than in iron given the same field strength and material volume, though in a transformer,
or generator, or alternator, these are never the same volume).
air gaps are typically very thin, compared to their cross section, they contain very little
volume, but this is measurable and the contribution to total energy is calculable, and
hence the division between energy in the iron core and the air gap is calculable, though
as DK points out this division ratio isn't really important.
but the air gap geometry, typically being thin with a large surface area, means the fringe
lines of magnetic field that bulge out around the gap are negligible, IE its not the distortion
of the field that matters.
it is my belief (I could be wrong here, might need some myth busting !) is that the reason
for the air gap in a flyback transformer is to avoid saturation of the core, so that its response
to the applied ampere-turns is more linear which makes the oscillator easier to design.
An additional reason is to get more energy per unit volume, reducing cost of materials (this
belief is backed up by google searches).
now back to CDI vs LDI, if DK needs to increase the inductance of the coil in order to
lengthen the spark's burn time in CDI, then an air gap in the coil is the wrong direction to
be looking since an air gap decreases the inductance of a coil (which can be compensated
for with more turns of wire, but then that multiplies how many turns needed in the secondary
so becomes a cost-of-materials, and overall size and weight of coil, issue).
I have played with flyback transformers, but have never designed the oscillator from scratch,
so I guess I need to do that some day to add to my knowledge of the mysteries of the universe.