test
Coil turns 76.5 turns , 151 turns , 224 turns
R is 0.068 ohm , 0.152 ohm, 0.25 ohm
Total induction no core across all turns is 1.49mH
Induction with core plus shoe is 9.25mH if the other core is placed 7 mm opposite, then induction is 10.23mH
Air core 151uH , 647uH, 1.53mH
core and shoe 1.1mH , 4.28mH, 9.2mH
oposing cores 1.23mH, 4.75mH, 10.2mH
R 0.06 , 0.15, 0.25 ohm.
According to this data, the longest L/R time constant is 40.8ms when the inductance is 10.2mH and resistance is 0.25Ω.
This analysis demonstrates that when the current flows in such LR circuit longer than 0.5757*40.8ms=23.5ms then more energy is dissipated as heat in the resistance than is stored as the magnetic field of the inductor. Heat is irreversibly lost energy.
When using the 5Ω rheostat + the commutated variable resistance 1+1+5+5+5+1+1Ω, the break-even time shortens proportionally (up to 96x shorter when all these resistances are in series).
Thus any hope for high-efficiency through electromagnetic induction and magnetic energy storage and recovery is dashed.

Clemente Figuera revisited
Reciprocating bucking magnetic flux of
2 coaxial solenoids energized by out-of-phase currents.
Clemente Figuera revisited
Other effects, like phenomena occurring in the arcing of the commutator are not so disqualified.
However there are so many arcing systems that have been built with a single contact or a single spark gap, that it is very implausible that multiple segment commutator would be any superior or involve a novel arc-based energy gain mechanism that a simpler single contact/gap did not. ...unless you are Rube Goldberg.
This points to the conclusion that the entire electromechanical commutator with the resistor ladder is just a slow low-resolution DAC and sequencer that modulates current flowing in these two sets of windings.
The way this DAC is connected points to an intention of creating a time-varying current that increases in one set of windings while decreasing in the other set of windings. Possibly to keep their sum constant ...and the fluxes that these currents generate, too.
Since any claims of energy efficiency of transformer-like induction are refuted due to dominating resistive losses at plausible frequencies, the remaining investigation of the gain mechanism should focus on the magnetic flux distribution (and its virtual movement) in the air gap of differently arranged windings. Not necessarily coaxially.