Lidmotor's Penny circuit help needed.

[Dark Alchemist]

500ns/div to zoom in on it.

[TinselKoala]

OK, so now you have changed the 800K resistor for one of much lower value, and you are getting much more conventional JT behaviour from the circuit. I presume it no longer does the "burst mode" but rather is lighting the LEDs continuously now?

The 500 ns shot shows clearly: the Ch A voltage rises to the combined fwd voltage of the 4 LEDs, which is around 13.5 or 14 volts. Then the LEDs begin conducting, the Ch B voltage (current) rises quickly to its peak value. Then the voltage drops when the system saturates and the current goes back to zero.
The peak voltage on the "current" trace is about 20 mV. So by Ohm's Law the peak current in this 0.1 ohm resistor is 0.020/0.1 == 200 mA. The LEDs are probably pretty bright. The voltage is pretty level at about 14 V during the ON time of the LEDs so your instantaneous power is around 2.8 Watts.
The duty cycle looks to be about 4 or 5 percent HI. So the average power is around 140 mW or about 35 mW per LED. If the LEDs are 20 mA, 3.45 VDC LEDs then they should be drawing twice that with DC power, or 69 mW each, or 0.276 Watts for all 4.
(3.45 x 4) x 0.020 == 0.276 Watts

So... is the stack of 4 LEDs powered by this JT as perceptually bright as they would be, if they were powered by straight DC at 69 mW each? That's the practical efficiency question. Is the output electrical power lower than or higher than the input power? That's the "overunity" question.

I think. It's early though and I may be suffering from a coffee deficiency.

There is one problem though. The 1.2 nF capacitor across the LEDs is part of the load, but your current measurement resistor is inside this loop, so the cap may provide a bypass of some current that isn't going through the 0.1 ohm CVR. You really should put the CVR outside the cap-LED loop (Move the bottom cap connection from the ground up to the other side of the CVR where the probe is). Will it still oscillate that way? I don't see why it wouldn't.  Is this cap really necessary? In my hardware build it doesn't seem to do much.

[Dark Alchemist]

Without the cap, and the cap value must change depending on how many LEDs there are, it will not oscillate.

By the way when I remove a LED both curves do change so it is following at least somewhat properly but can I can't get more than about 250ma out of this circuit.

Here is a screenshot with the resistor outside of both the LEDs and the capacitor and now we seem to have a huge negative spike on ch. b.

[Dark Alchemist]

I can't seem to get rid of that huge negative current spike.

[TinselKoala]

Well, I'm still trying to get the proper component values in my construction. I wound a toroid today with the right inductance, anyway, and tomorrow I'll check with my supplier to see if I can locate a BC337 transistor.
But now, with this new toroid, I get two modes of operation. One is the true JT mode at around 60 kHz and the other is ... weird. Dimmer lights and much lower frequency and a different waveshape. To get the true JT oscs I do need the 1.2 nF capacitor; if I remove it it shifts to the other mode. To get the true JT mode I have to do something to the battery connections, introduce noise or something, I'm not quite sure what I'm doing to get it.

I'll make a video later on this evening. But here's the toroid inductor, measuring one winding. The other is the same.