[In what follows, I will agree to your request and use L rather than l for length symbol, and further use V rather than v for volts]
Quote by PeterDonis
Qreeus: "Yes there can be an E > Ecrit in some other frame 'immediately upon switchon', but that alone is *not* sufficient."
Of course not. I was not arguing that it was.

Really? Go read my quote from your #321 again. Yes it is what you were arguing. And even more clearly so in your #352:
"And taken to its logical conclusion, this would again imply that breakdown should occur *immediately* upon turning on any field source, regardless of its state of motion, since there will always be *some* vp that will sense E > E_crit. That's obviously false, so again something must be wrong with your argument."
Up to you whether to admit that claim, made twice, is mistaken.
Qreeus: "It must be sustained for a minimum period of time, seen in that frame."
As far as I can tell, by "that frame" you mean *any* frame at all. Correct?

Provided that frame, in which E >= E
_{crit}, relates to motion wrt to an E source meeting invariant E*L >= ~ 10
^{4}V/cm.cm (E and L being orthogonal). As I have consistently maintained, two criteria must be met.
Qreeus: "That in turn imposes minimal restraints on the combination of size and applied E of the E source, which I gave before as E*L >= ~ 104V/cm.cm, true for any frame whatsoever."
So, cancelling the units, we have E * L >= V_crit, since the units of the product are now volts.

Only nominally 'volts', as clearly explained in #356: "there is an invariant E*L product (E and source characteristic length L being orthogonal) that must be exceeded before breakdown is possible in *any* other inertial frame." Further, in #360: "I am claiming, based on example in #338 and codified in #356 as E*L >= 'volts'
_{min} (even though it is not really volts per se)" and later; "Putting it more concretely, from #338 example, a capacitor must roughly have M = E*L>= 10
^{4} (V/cm)cm before a vp pair passing through at any relative gamma factor whatsoever can be elevated to real pair status." Notice I expressed it variously as either (V/cm)cm, or V/cm.cm, in order to avoid any confusion that the quantity is really 'volts'.
Rereading #338, where this scenario began, it's more than clear E and L are to be taken as orthogonal in order to make any sense at all. And as quoted above, that criteria was expressly stated in #356. Forget it all somehow?
And L is some characteristic length associated with the source, such as the distance between the capacitor plates, correct?

Clear by now surely that L is
normal to applied E. And btw, just as clearly, relative velocity (re gamma factor) is also taken as normal to E, hence along the same axis as L. That was also explicit and implicit in #338 and later. Otherwise, frame dependent E coupled to frame dependent duration would make no sense.
And is L supposed to be measured in the source rest frame? That seems the most natural interpretation, but please confirm.

It's the most natural frame, but the invariant product (let's express it now as ExL, even though it's not a cross product) is all that matters here. Choose any frame you like, so long as E and orthogonal L, measured in that same frame, give a scalar product >= ~ 10
^{4}V/cm.cm, which is *not* to be taken as volts.
Qreeus: "Further, as E >= Ecrit must hold in the rest frame of any chosen vp pair, the additional constraint follows that, in E source rest frame, gamma factor of vp pair passing through is gamma >= Ecrit/E."
How do I measure gamma? After all, as you agree above, I can always find *some* frame in which E_crit / E is greater than any gamma value I choose. So if I measure E in a given frame, and take the ratio (since E_crit is known in terms of physical constants, I don't have to measure it), how do I tell if your criterion is met? What gamma do I compare it to? I see a possible answer below, but it would be nice, once again, to have confirmation.

Huh? The 'possible answer' is directly given in what you quoted above! It's the second criteria, as stated, given first criteria is already met.
Qreeus: "Any frame consistent with above. Recall I have said that invariant vacuum vp spectrum demands a finite ultrarelativistic flux of vp's in a source rest frame. Provided source parameters meets minimal impulse criteria E*L >= ~ 10^{4}V/cm.cm, one simply singles out for attention *any* vp pair having sufficient gamma factor wrt source rest frame such that transformed into that vp pair's rest frame, E >= Ecrit applies."

How much more explicit can I get!?
Hmm...so it looks like you are suggesting *two* criteria:
(1) In the source rest frame, E * L >= V_crit.

Actually, that's ExL >= ~ 10
^{4}V/cm.cm, in any frame whatsoever, and understood that V/cm.cm does not mean volts.
(2) In the rest frame of a pair being created, gamma > gamma_crit. where gamma_crit is the gamma necessary to make E >= E_crit in the pair's frame. given that the first criterion is satisfied in the source rest frame (i.e., gamma_crit is calculated relative to the source rest frame).

Congrats  got that bit right.
My first comment is that I don't understand why the second criterion is even necessary; again, given that the first criterion is satisfied, there will always be *some* frame in which E >= E_crit, so I just figure out what the gamma is for that frame relative to the source rest frame, and say that that's the frame in which pairs will be created at rest. So if the first criterion is satisfied, the second must always be satisfied in some frame. Why then is the second criterion necessary?

Because, as discussed at length above, you got the first criteria wrong. The two criteria must be simultaneously met. Which amounts to nothing more than saying an E source must transform by the LT's into an applied E >= E
_{crit} in any vp's rest frame, for a minimum period of time in that frame given by HUP. For any given E value in say source rest frame, L minimum is thus set there. And vice versa. Minimum gamma factor for any vp's whizzing through in source rest frame is then set solely by E there according to gamma >= E
_{crit}/E. A no brainer combo.
I can see why you might want to use the second formula to predict, for example, what the initial current due to the virtual pairs would be in the source rest frame (since that will depend on the velocity of the created pairs), but why is it a criterion for determining whether breakdown can occur at all, given that the first criterion is satisfied?

The answer should by now be selfevident. And btw, I take it we agree transverse velocity of a created pair is an overall neutral mass flow, effecting the current along E only insofar as 'relativistic mass' of pair will be very high, making motion along E extremely sluggish as seen in source rest frame. In fact, as per discussion in #360, it implies many created pairs simply whiz right through without ever being collected by E source.
My second comment is that I can always meet the first criterion by making my capacitor plate separation large enough, given some limit on the E field I can produce at the plates.

And that error of understanding what L must be referring to invalidates the rest of your critique. Still, there is an issue regardless as per #360. While it's not simply a voltage V, but an an variant product ExL >= ~ 10
^{4}V/cm.cm, it can be easily enough met. So the real task, once all the basic misunderstandings above are finally cleared away, is to figure out what is really going on and why. So far, no order of magnitude estimate of expected pair flux has been made. Could be typically so small as to go unnoticed. If QFT already has a pat answer making that moot, noone here has so far offered it.
Assuming no pair creation near hightension power lines etc., one conjecture might be to suppose that applied E somehow suppresses high gamma factor flux of vp's in source rest frame. Immediate problem with that is for a capacitor configuration, many high gammma vp's would be originating outside of appreciable applied E region, so subsequent reduction to gamma < gamma
_{crit} within E region implies weird electrodynamics indeed.