In the process of perfecting chainline by moving the rear axle to the right, what is the least amount of axle that can be protruding from the left dropout while remaining safe to ride? In other words, does the nut have to fully wrap the remaining portion of axle or would be say 3/4 be enough to get sufficient tension? Also--if it matters--the dropouts are near vertical.

Respace the axle so you have enough to put the entire bolt on.

+1, there is no sense in riding a cheesy dangerous setup to save ten minutes of extra time and getting the job done right.

Not an option, I'm afraid. The chainring cannot be brought any closer to the centerline w/o hitting the chainstay. If I shift the axle to the left any at all it compromises the chainline. It is a give and take at this point, and I'm just curious how much I can get away with.

Move the cones leftwards. I've never done this but they are moveable right? Sorry if i have no idea what im talkinfg about, there is a good possibility that this is the case.

No they are machined into the axle.

D'oh

cartridge bearings with stops that are machined onto the axle itself.

Right, right. Well I'm stumped. Back to your original question, it might be safe to ride like that but it goves me a bad feeling and I wouldn't do it. Someone with more engineering knowledge than me will have a more objective/ scientific idea of how safe it would be.

I can tell you this much. From many years of taking wheels on and off, it "feels" like it has enough tension. What I'm concern about is will the threads hold up? I've been riding the bike for a couple of hundred miles now without incident, I'm just paranoid about the setup. I don't like to ignore that little voice in my head if you know what I mean.

I think 3 full turns of the nut is the general recommendation for a lot of stuff...I'd never let the axle go with 3, but if you are screwing what I'm assuming is a normal sized track nut on there, 3/4 might be plenty. Take it off and put it back on, counting how many full rotations it makes. I'd say it's probably fine, especially since it's on the non-drive side, but I don't promise

Well, the tension is set by how hard you're cranking on the nut. If you're not shearing the threads when you tighten the wheel down, then you're probably good.

I'd say 3/4 of the nut (assuming it's a big fat one like the ones I use to hold my wheels on) is more than enough.

Well the threads certainly aren't sheared, and I would say that the tension is definitely greater than it would be with a quick release if I were using one. The problem really stems from the design of the nuts themselves. The nut has one of those integral washers--the kind that lets the nut spin without digging into the finish of the bike itself. If I were to replace it with a nut such as the "fat one" you describe without a washer I would probably be able to completely thread the nut. Someone remind me what is the size and thread I would need to go hut for at the store.

You want the nuts you have on there now.

Yeah, I decided the track nuts would hold the best too. I'm just going to ride and take my chances; being on the non-drive side whats the worst that could happen?

You could also think about putting some blue loctite on it if you're worried, but if you do it you'll probably want ti carry a longer wrench around in case you get a flat.