I think it more likely that after 2 turns you need to discharge them completely or risk a catastrophic discharge, that is they explode, doing 36EP per ton of damage, inside your ship.
However, the problem with this interpretation is that it completely wrecks the mechanics for how black globes work in dumping energy into and out of capacitors...
It was an off the cuff idea

(it needed as I noted more fact checking)
But I think you hit on a key between that above and this...
Let's do a little handwavin...
I am confused here -- why do we want to do this again? Why do we not just want to use the things as originally written?
Someone else said about this issue once (maybe it was Whipsnade?) that the point is exactly that. To paraphrase from my poor memory, the only way and reason jump drive capacitors can be used in that way is because they are hooked up to a black globe generator. They just can't be used as batteries without a black globe generator. That is using the things "as originally written".
...which is the entire reason that capacitors get mentioned in HG in the first place.
Not exactly if I read you correctly. The entire reason they get mentioned is for use with black globes. Only. No other usage.
The whole challenge with using capacitors as a sink for energy that strikes the black globe is precisely the fact that the energy does not dissipate over time and must be otherwise expended at a quantified rate.
True, to a point at least. We know the jump drive capacitors do not lose charge, at least not at an appreciable rate for the purposes of HG combat. But most battles will be over in a few turns. And battles involving black globes will require judicial monitoring or you explode. Destroyed. Most of the time dumping the power absorbed is not a problem. Personally I prefer a fighting chance, hence the 36EP per ton internal explosion. Sure, that's probably the same thing...
We also know that if a ship cannot charge the capacitors in the jump drive within two turns it cannot jump. Why is that?
Significant capacitor leakage I'd say. Such that the leakage is enough that in the third turn the capacitors have dropped half the charge applied. I expect in the normal operation of ships this falls within the safety limits of being able to discharge that energy through the ship's jump drive as you note. Or maybe that charge isn't even lost, just dissipated through the rest of the capacitor to the point where it can't serve to imprint the full charge for jump. Flush the system and start again.
Also note in HG "Energy used to power the jump may not be used for other purposes." So it is just waste energy dumped by the jump drive if not used to jump, the way I see it. Without a black globe generator to allow repurposing of the energy it would simply fry the ship's circuits seems the interpretation.
Before the handwavium about hull jump networks was introduced, the 2-turns-worth-of-EP rationale was "lanthanum coil charge reduction" -- it is not the capacitors that leak charge, it is the other, time-and-space-bending bits in the j-drive, which actually create the jump, that leak energy away (probably into j-space or somewhere) while energized but not engaged.
I'm ignoring the hull jump networks (you're talking grid right), I don't care for it, more problems raised and ignored than settled. Where is the bit about coil charge reduction from?
Note also that the default EP capacity of a j-drive's capacitors is far in excess of two turns' out from a PP rated for the j-drive's maximum jump number...
Quite, which is why you need all that fuel burned as well to jump. That is the other part of charging the capacitors. The initial bit of two turns of powerplant output is just to smooth the charge, imprint them if you will, with the desired energy pattern to create the jump you want.
...and without going to town about the Annic Nova, suffice it to say that really, really big capacitor banks, charged over a period of many days, can even eliminate the need for jump fuel entirely.
Umm, no, they can't. Not if you ignore the Annic Nova (alien tech) and use the rules as written "If a ship absorbs enough energy to make a jump,
and is supplied with sufficient fuel, it may jump at the end of the turn." The interpretation there seems to be the "enough energy" is two turns worth of powerplant output, and the "sufficient fuel" is the normal fuel required to jump, which is used to create the quick charge over the imprint. The implication being slow charging the capacitors does not work for jump. Ignoring the Annic Nova (alien tech) exception.
In practice, the convention always seemed to me to be that the j-drive capacitors were there to provide a reservoir to help manage the energies required to initiate jump and subsequently accommodate excess energy produced by jumping and burning all that fuel so quickly... in addition to allowing starships to bank enough charge ahead of time -- and maintain it in a ready state indefinitely -- to jump on a moment's notice...
I just dont see the banking charge and moment's notice bits in the rules.
And I'm losing my train of thought and focus here so I'd best wrap it up and correct any errors later
