We have read the patent and at first glance don't buy the discussion & claims.
First off, the incremental inductance is proportional to conductor length, and then the outer turns contribute more inductance than the inner turns. The coil supports the applied voltage (assumed AC here), thus the outer turns with have a larger voltage drop than the inner turns with the sum equal to the applied voltage. This will cause the voltage difference between turns to be non-uniform and larger on the outer turns and smaller on the inner turns, this alone seems to violate the patent discussion starting at ~85.
Secondly, hopefully the coil isn't an "Over Unity Device", so storing more than "250,000" times as much energy would imply that a simple 1H coil with applied 100VAC, assuming 60Hz, would conduct 265ma and store 0.035 Joules, so this "specially wound" coil would store 250,000 times as much energy as a conventional coil, or 8,795 Joules!!!
We must remember that inductive coil energy is E = (L*I^2)/2, and this implies a coil inductance of 250,000 times increase, or 250,000H that's just not going to happen!!!
Applying EM analysis to the coil described in patent 512,340 and one should find some interesting characteristics, one of which it doesn't store 250,000 times as much energy as a conventional coil!! For simplicity think of the incremental inductance per coil turn and calculate the incremental stored energy per turn, as the sum total should be equal to (Leff I^2)/2 where Leff is the total incremental inductances summed over the whole coil.
Also from First off above, the incremental inductance concept is utilized in special wide-band RF coils commonly used a chokes where the coil is wound in a cone shaped spiral. This unique shape gives a much wider inductance range over frequency than a simple solenoid or flat spiral type coil. A little EM analysis will show such.
You could easily construct a small coil for physical measurements. Please be careful as if we are wrong one could easily kill themselves!!
Good luck on your intellectual endeavors!!
Best,