Hurr, sorry. My only bike is in my sig, which is an '82 CB650. I know it's not the breather, since that was still firmly attached. Mystery hose runs down through the center of the bike. I'm beginning to suspect it might be some kind of fuel overflow hose for the carbs, or as was suggested, a balancer of some kind.
Hush, if you think the dust is bad, you should see the gunk in there. I have what I thought was the infamous head gasket leak, but I cleaned up engine a bit yesterday and today it appears that the leak is coming from the top of the engine where those valve covers sit. I'm relieved, since I imagine it will be much easier replacing that gasket than removing the head. But the engine is in dire need of a really good scrubbing with simple green. I think this winter I will be taking the engine out and going over the whole thing. Lord help me.