Looking at some large pot cores , I was struck by this novel idea.

Back in the very early days of 10GHz with wideband equipment, we tuned Gunn oscillator using a dielectric probe on a micrometer head - that allowed precise adjustment with a repeatable reading.    Looking at my gapped pot cores, it occurred to me something similar could be done with them.

Fix, somehow, one half core to the micrometer so it can me pulled away from the other allowing a precise air gap.  Of course, doing it this way means you have a gap in outer and inner pieces, so the effective gap is double the measured value

Inductance is then uo * A / g    where uo is mag permaeability of free space, 4E-7 * pi   A is the core area and g the adjustable gap  (SI units) 

You'd have to watch out for saturation (V= 4.44 . N.A.B) but as a small adjuster to a larger fixed loading coil at low frequencies,  may have some merit for fine repeatable setting

Just a thought

Andy  G4JNT