There is a puzzle. The measurements and the waveform show absolutely NO energy being delivered by the battery when the gate voltage is positive. Something is blocking that current flow. As you've pointed out it's probably due to the offset. Remember that we have 6 x 12 volt batteries being applied during the 'on' time. That's a pretty hefty kick and there's a clean voltage waveform across the gate.
The thing is that when it then goes into 'burst oscillation mode' as you put it - then there is absolutely NO restriction to the flow of energy from the battery. That is also evident in the waveforms and in the voltages measured. But if the battery is then delivering current from an induced voltage it is also getting it back. Then again. Nota bene Pointy. If we disconnect the power for a period of 3 minutes - there is a dramatic drop in the heat measured at the element. But that 'oscillation burst', conversely, is sustaining that temperature and even increasing it. Nor does it decay. Not even by a fraction. The only time that there is clearly some distortion to that waveform is when we apply a switching frequency in the Megahertz range. Otherwsie it remains - as steady as a rock - evidently doing some useful work. And it is NOT NOISE. You can see it for yourself.
Also. It then appears to return more energy from the circuit than was applied. To my way of thinking that means that there is more potential difference induced in all those circuit components than was delivered as current flow from the battery source. For some reason - which I simply cannot understand - you require us to reduce the amount of wiring. Why? If this is all adding to that inductance? The wires used are only as long as is required to reach the circuit apparatus and its sundry components. And there is no question that there's inductance over the heat sinks and the resistor itself. (Still to be measured by the way. Hopefully tomorrow.) There's potential for inductance all over the place. But nothing out of the ordinary has been added. It's not 'smoke and mirrors' as Humbugger claims.
Another point. The battery voltage is CERTAINLY showing those changing values. Not as quick as the scope shows us - but the DMM used cannot operate at the same frequencies. But it moves up and down - just as one sees when one's recharging a battery through a standard recharger. And at some settings that variation can be extreme. And the DMM IS precisely at the positive and negative of the supply.
So. Here's my question. If the circuit requires a certain level of inductance in the element - or even a certain size heat sink at the transistor - or even long wires, which in any event is an inevitable requirement on any application - and if all this is adding some critical value to that 'negative' potential - then how does eliminating them disprove their value? Unlike usual applications - this particular example is intended to exploit that 'burst oscillation'. It clearly depends on induced voltages in circuit material. And hopefully, we'll find a way to test this with even longer periods between each switch. I think what's needed is some way of initiating the first trigger to get it to oscillate. And then just sit tight and let it do it's thing? It probably sounds preposterous. But the indications are that it will just keep on keeping on. We need to find out how long it can manage this.
Then. Regarding the measurements. You must appreciate by now, surely, that our measurements are about as precise as can be managed. Our DSO's are top of the range and - fully calibrated - they carry the manufacturers' ratings. So. When it shows a voltage reading - I think one can pretty well bank it. Otherwise one must give up on measurements altogether. I have no idea what's happening at the battery. I simply cannot tell you if it's discharging or recharging or even sustaining a charge. But I can certainly advise you and all classicists that - on the face of it - there's evidence of a very real potential - thus far overlooked. I've said this before. We just need to take an average AC supply - channel the positive to one application - the negative to another - and then send all that extra energy back to the plug and to the grid supply. Not that this is realistic. Just that this is the logical consequence of where this result is pointing.
In any event. This is CERTAINLY one of the 'effects' that we hope to demonstrate. I'd be much more encouraged to actually see the question answered after the required research. All we can do is show it experimentally. And I must add this. There is evidently some very real and exploitable energy in all that negative potential.