If you're at five tenths across the ways, you're probably a lot more level than most lathes are. That's not enough to explain 4 thousandths in eight inches.
Actual level doesn't matter here. You want that, but using a level that's five tenths per division, no one, even you would know the difference if you shimmed up the low leg, took a shim out of the high leg, or shimmed the opposite side of the lathe so that both ends were out equally. That's really dang close.
Once you've got a part in the chuck (any chuck), or a collet, or hanging off of a face plate, or welded to the spindle if you wanted to, (don't do that), the part may or may not run true. After you take a pass or two along the part, (or take a facing cut), that's all irrelevent. You have reduced the headstock errors to ONLY the spindle bearings. Even if the spindle is bent or twisted, you've still eliminated that. The part, whether it was straight or not, only rotates about the axis formed by the headstock bearings. So.... Don't worry about any of those items yet.
With eight inches of stickout, did you have the part supported on a center? What was the diameter of that test piece? Is it a solid piece, or did you do a two collar type?
When you faced the piece of aluminum, when you had the indicator on it, did you indicate ALL the way across it, or only to the center? And indicator will read zero on the entire first half, from the near side to the center, it's the back half that will show the error.