I'm not entirely sure that a resonant condition is what is wanted in order to get more efficiency or higher output from a JT-type circuit. After all, a resonant tank stores energy, and as the tank components aren't perfect, they will inevitably be dissipating some of that stored energy in unwanted ways, like by Joule heating of the components and RF radiation, removing it from being available for "output".

It might be true that a resonant condition could enable a circuit to extract some energy from the "ambiance", like a tuned receiver of the electrosmog harvester or crystal radio type does, allowing some of this "outside" energy to be put to use by the circuit.

The old JT threads on this forum have a lot of information as to making JTs as efficient as possible by carefully tuning coils and circuitry. The closest thing to a "resonant JT" that I can think of is the Slayer Exciter-type solid-state Tesla coil kind of thing.

Resonant switching reduces switching in the switching transistor(s) and diode (if present).  An archetypical JT uses regenerative feedback to speed up switching edges, but still suffers losses due to switching the transistor off when the inductor current is at a maximum.  The "cool Joule" has even more switching loss as the switching is not regenerative.

This is exactly the problem he had.... no one would listen to what the was saying, which is based on simple, well known inductor theory...
He evne built a 9-inch torroid and tried to show people, no one could figure out why it did what it did, despite his attempts to show it to them..
he eventually sold his patents and moved to L.A.

Where he buys his power from the local grid just like everybody else.

Now, as we have determined, the "cool joule" or what I'm calling the TMLMJT circuit we have been discussing DOES depend on the resonant tank formed by the L1 coil and the Base-Emitter capacitance of the transistor, and operates at the resonant frequency of that tank circuit. I've just done a measurement of the tank resonance using the setup pictured below, by sweeping the FG's sine wave output and reading the voltage response of the tank, looking for the maximum p-p voltage, then reading that frequency using the Philips counter. The value is in agreement with the power-on operating frequency of the circuit.

Now, as we have determined, the "cool joule" or what I'm calling the TMLMJT circuit we have been discussing DOES depend on the resonant tank formed by the L1 coil and the Base-Emitter capacitance of the transistor, and operates at the resonant frequency of that tank circuit. I've just done a measurement of the tank resonance using the setup pictured below, by sweeping the FG's sine wave output and reading the voltage response of the tank, looking for the maximum p-p voltage, then reading that frequency using the Philips counter. The value is in agreement with the power-on operating frequency of the circuit.

can you loop this back to recharge a set of batteries?

When you get the timing right, as soon as the transistor cuts off - the field collapse induces a current in the opposite direction, and just as that stops, a pulse comes from the transistor again.

Again:  The archetypical Joule thief circuit does not have a resonant tank.  The on time varies with the saturation characteristics of the transformer and the battery voltage.

In Resonance::

I will use "transformer", because it is center-tapped bifilar inductor, which makes it a "transformer"...
[The signal through the transformer is no longer a function of pulsed DC, but rather an A/C waveform, of energization and field collapse in rhythmic pattern, as visible on the scope. the energy from the battery is being converted to a higher voltage and the current changes over time, until the LED turns on. Since the resonant frequency is (100Khz-xMhz) faster than the response time of the diode, the LED does not turn off.]

All oscillating circuits are "rhythmic".  Certain oscillators use resonant tanks.  Many oscillators do not use resonant tanks:  relaxation oscillators, RC phase shift oscillators, and blocking oscillators are all examples of oscillators that do not employ resonant tanks.

Flip Switch:
Current flows through base and voltage is stepped up until it reaches cut-on potential::

Wrong.  When base current flows, the transistor turns on charging the inductor with more current.  At this time the collector voltage is close to zero.

Transistor on: the Inductor is being energized - current flow is not linear, but a function of COS, change in voltage over time is a function of the inductance times the time-variant current flow. In resonance, this is not linear either. Voltage increases until the LED turns ON.

Once again, the collector voltage is very small during this phase of the cycle.  When the transistor turns-off, the inductor flys back rapidly increasing the collector voltage.

Transistor switches off:  Current reverses direction through the inductor as the magnetic field collapses.

That is absolutely wrong. The collector voltage increases because the inductor current continues in the same direction.  The transistor turning off blocks the path to the emitter so the collector voltage rises as the inductor current charges local capacitance until another lower impedance path conducts.  The LED provides that path once the collector voltage reaches the LEDs V_FW.  The collector voltage waveform approximates a trapezoid, not a sine wave.

The other half of the sinewave presents itself across the coils, since there is a diode, it only exits out the secondary.  Which makes a connection to both the battery and the base resistor. The induced voltage, and the time-variant current flows through the resistor until it reaches cut-on potential and the transistor turns on again.

Rising collector voltage induces negative going base voltage in the base side winding.  The transistor is held hard-off by the winding until the magnetic field in the inductor diminishes to zero, elminating the BEMF that works against the battery voltage, and the net value holds the base off.  The battery then once again forward biases the base-emitter junction, the transistor begins to conduct and the falling collector voltage induces BEMF in the base winding that increases base drive, regeneratively turning the transistor on.

The LED has not yet stopped emitting photons.

Perhaps if you have LEDs manufactured in 1973.  Not so much for LEDs manufactured since the turn of this century.

By the time the inductor runs out of energy, the transistor is on again, recharging it. <- if not, the system is NOT in resonance.

The system is never in resonance.

Transistor On Again: remaining voltage flowing from secondary coil + battery recharges inductor, and the cycle repeats itself. Adding to the voltage each time, until it reaches system maximum.

After a number of cycles, the system reaches an equilibrium.

a bunch of LEDs will discharge it more quickly, but also take longer to charge the inductor. and the voltage drop across each diode affects the total voltage over time induced in the coils.
 You will notice each diode you add, they all (except maybe the first one, depending on the type of transistor you use) will get dimmer and dimmer, until no more of them light up at all. Your circuit may handle 10, 20, maybe 40, but eventually you will reach its' potential.

In the archetypical Joule Thief, the transformer swings through 1/2 the magnetization curve. During the transistor on-time the transformer goes from zero bias to saturation.  During the transistor off-time it returns from saturation to zero bias.  The energy that the transformer stores and discharges each cycle is fixed by the transformer saturation magnetization energy.  More series LEDs decreases the transistor off-time, increasing frequency, and thereby increasing the power stored and released by the transformer.  However, it also increases the switching loss of the transistor.

The mechanism for resonance is not "real", its simulated, by the switching of the transistor in place of where a capacitor would be in a resonant LRC. The inductor doesn't know the difference.

Wrong again.  An inductor in a resonant tank operates on both sides of the B-H curve.  In the archetypical Joule thief circuit it operates on only one side.[qutoe]

Instead of replacing the transformer,  you can take your transistor out and replace it with a 555, if you set it to switch at the resonant frequency.
The inductor doesn't know the difference.  although the 555 has its' own internal capacitance, so this will change the resonant frequency slightly.[/quote]The capacitance of the 555 itself as well as the capacitance of the timing capacitor on the 555 has no bearing on the resonance of an external tank circuit, which again:  the archetypical Joule thief circuit does not have.

It applies if you apply it.

can you loop this back to recharge a set of batteries?

Yes
When the transistor becomes open circuit(switches off),a current loop is created between L2,the LED and B1. What power isnt consumed by L2 and the LED in the form of heat and light, charges B1. You may put as many batteries in parallel as you like in the B1 position,and they will charge. But because of switching losses,heat and light output,the charge returned to B1 will always be less than that supplied to the system by B2

Tinman this is the comparative result with and without a base resistor with no load.  The flyback voltage increases about 15%.