This is getting rather ridiculous, don't you think?

Are you really looking for, and trying to reproduce, some kind of OU evidence that only shows up with the cheapest possible DSOs used incorrectly? With absolutely NO concurrent validation (a technical term meaning evidence of OU from some OTHER experimental or measurement technique running alongside the Atten measurements)?

Note again, that after all this time, the definitive experiment is still not being done. Charge a known capacitor to a known voltage. Then you know precisely the energy available for whatever. Use this capacitor to run the JT circuit while monitoring its voltage and the time accurately. Monitor the output power in your favourite manner. When the capacitor has dropped by a known voltage increment, you will KNOW just how much energy you have put into your circuit, and you will KNOW for how long.... thus you will KNOW, not guess or interpret, just how much input ENERGY you are dealing with during that time. This figure can be directly compared to the output power, or more precisely to the integrated output power over the time interval concerned (the output ENERGY).

It will do no one any good if I do this measurement. It must be done on several boards that Lawrence claims do show OU behaviour, preferably by Lawrence himself, but anyone who has an "approved" board, an oscilloscope, a stopwatch, an accurate voltmeter, and a bunch of capacitors can do the work in a day or two with enough runs for statistical significance to result IF there is anything to see.

When you see that every capacitor you try results in efficiencies of under 100 percent, then you can start resorting to your special pleading that "the robust and confirmed OU effect only happens with batteries. No, not those batteries, or those, not just any batteries, but only rechargeable batteries of specific chemistry and manufacturer, and only when they are between 0.341 and 0.351 volts".... and only when they are pointed northeast, on Tuesday mornings, when the weather is clear......

Whatever, dudes.

Relax Tinselkoala. You are not stating anything that is not already obvious to me, anyway. At this point I have agreed to test a couple of Lawrence's boards to see what shows up in my own measurements. I make no assumptions at all in regards to what is happening in Lawrence's boards. Once I get a couple of boards to test with, I will make measurements and see if there is anything unusual going on there. I will report all my results back here anyway. Anyone who has any concerns with my measurements will be free to provide whatever feedback they like. The world is still safe.

Lawrence, I think I see how you may be getting that ringing now on your input current waveform. I connected a super capacitor charged to just over 0.4V as the input voltage source for my joule thief circuit, and at 408mV input voltage I am starting to see some ringing on the input current waveform with a little bit of zero crossing. This sort of ringing on the current waveform would not be unusual in itself when you are driving an inductive load with a capacitor. The combination of capacitance and inductance can certainly produce ringing, so that in itself certainly does not imply anything unusual or imply over unity. However, it will be interesting to conduct tests to see if this ringing does reduce power consumption or increase efficiency at all, and if so, by how much.

I have attached a screen shot of the ringing on the input current waveform (blue trace) that I am seeing using a capacitor as my DC power source for my joule thief circuit. The capacitor is currently at 408mV. You can see a little bit of zero crossing on the larger negative peaks of the ringing pattern. Again, the ringing pattern seen would not be unusual in a circuit with capacitance and inductance, and the zero crossing certainly doesn't necessarily imply that there would be an increase in efficiency or over unity. Measurements of input and output power at different input voltages with a capacitor used as the input voltage source could be used to see if efficiency increases by any measurable amount when this sort of ringing with zero crossing occurs at some given input voltage, as opposed to say using a regulated DC power supply set to the same input voltage, if no ringing occurs on the input current waveform when using the regulated power supply. If possible, I will try to run some tests of this sort to see if the ringing has any measurable effect on performance.

@Void
 
The Zhou Boards showed much ringing - heard by the many young ears.  (I could not hear any but that was probably due to my poor hearing.)  I shall use different time scale to show the ringing waveforms.
 
I noticed that the display of your DSO is exactly like Atten.  That will make our communication much easier.  You can use the save/recall button to save the CSV file on the USB.  I can analyze one of your measurements for you (both Input and Output) as sample.  You do not need to rely on Vrms values any more.  You can use the vigorous Instantaneous Values.

Yes, your boards seem to show a lot more ringing than my circuit, and they also show some unusual pulses/noise superimposed on the waveforms as well, which may be just due to electrical noise of some sort, but I don't know. It might be something else causing those pulses. My super capacitor is actually 3000 Farads nominal value, so it is going to take a fair bit of time for the input voltage to drop down much below 0.4 volts. It is the only super capacitor I had available however. My joule thief circuit has been running all day and the input voltage is still at 400mV from 408mV yesterday evening. It might be a day or two before the voltage has dropped down much below 400mV. The ringing that can often be heard on a joule thief circuit is generally due to the frequency of oscillation being in the audio frequency range, which causes the ferrite to ring at that frequency for some reason, and probably not so much to do with the type of ringing pattern that can be seen on the input current waveform. My joule thief circuit seems to operate mainly above the audio frequency range, at least at the lower input voltages I have been testing it at, so I don't hear any ringing noise from my ferrite toroid. I will see if I can get the data logging working on the weekend. If I can do it using a USB flash drive, I would prefer that because hooking up to a laptop will likely increase the electrical noise in the test circuit. Laptops and their switching power supplies tend to produce a fair bit of electrical noise.

Analyzing the Noise.  Look at the following 5 slides.  The noise is relatively systematic.
 
What can it be???

Here is a scope shot of the input waveforms of my joule thief circuit with the super cap now at about 384mV.
The input current waveform still looks fairly similar to the way it looked at 408mV, but the frequency
slowly increases as the input voltage drops. The frequency is now around 294kHz at this input voltage.

Lawrence, it should be clear to you by now that the term "ringing" has two different meanings. First, the common meaning, an audible tone like a high-pitched bell ringing, that your students with young ears can hear. This is caused by loose windings on the toroid and perhaps the toroid itself, vibrating at the frequency of oscillation of the JT circuit, which as we know is a function of voltage as well as the toroid and transistor characteristics. When the oscillation frequency is in the range of human hearing you hear the parts vibrating.
The second, more technical meaning is the _inductive ringdown_ and this can be seen very clearly in your last, highest-speed scopeshot. This is the "bounce" in a signal that is seen when some energy is sloshing  back and forth between an inductor and a capacitance. As you can see, I hope, from careful examination of that last scope shot, this energy sloshes back and forth (the ringing) and dissipates (the decrease in amplitude of the cycles during each ring) but most importantly: IT AVERAGES TO ZERO, or slightly below zero since there is always a little bit of current flowing in addition to the ring. The frequency of this inductive ringdown is very high, as you can see, and is clearly out of the range of hearing. The inductances and capacitances participating in this high-frequency ringing are simply the wires and spacings in your circuit. This is why tight construction and short wires are used in high frequency circuit layout: to reduce this kind of ringing noise, which usually is considered detrimental to circuit performance.
Another feature to note from your series of shots is the difference between the first and second shots. You can clearly see that the first shot is missing a lot of the action, due to the "picket fence" effect of sampling and display rates and resolutions.  The first shot looks like random noise. The second shot is beginning to resolve the transistor's switching frequency (the potentially audible ringing). By the time you get to the last shot you are able to resolve the ringing of the inductances and capacitances of the wiring and layout themselves, far above the range of human hearing.... and acting as a purely dissipative mechanism for heating up the components concerned, as you can see from the fact that each cluster of ring oscillations decreases in amplitude over time.

(And you _still_ aren't putting your baselines directly on a graticule marker. See the little arrows on the left side of the display? Please use your vertical positioning knobs to put these little arrows _exactly_ on one of the horizontal graticule lines, unless there is a specific and clear reason for not doing so. You have no such reason, and the lack of accuracy in your scope setup is making it unduly difficult for observers to interpret your scope traces. I have mentioned this issue to you several times now. The reason for those wiggly lines on a scope display is so that they can be interpreted. The numbers in boxes that you seem to rely upon do not tell the whole, or even the most important story that an oscilloscope can tell. Proper management of the display of the traces is an essential part of scoposcopy. PLEASE PUT THE CHANNEL BASELINES EXACTLY ON HORIZONTAL GRATICULE LINES in all your future scopeshots unless you have a good, conscious and articulatable reason for not doing so. But congratulations on setting your trigger a bit better.)

If the "ringing pulse" represents current crossing the 0 ref line, then we must have positive and negative power in the circuit.  (Input Volotage is all positive.  Output Voltage and Output Current are all positive.)
 
What does that mean?  How can current flow forwards and backwards in a DC environment?
 
*** Look at the theoretical explanation again.  If there is a "hidden" pulsing source, the above is possible.  That is Lead-out energy.  So long as there is crossing of the 0 reference line - either the main Current or the spikes, there is lead-out energy.  Amen.
 
*** It is a matter of producing the circuit to achieve this crossing the 0 ref line behavior.  The Zhou Board is only one example.