New guy here, but I might as well jump in and embarrass myself right away. Because I don't yet know you guys, my apologies if my suggestions are things you've already done or thought about or just too basic. But alignment problems are usually harder to diagnose than you think they should be. I had the same problem with my Grizzly 5" chuck and was able to improve it noticeably.
First off, I am assuming you haven't had a bad crash on the machine recently, I know that only happens to other people. Also, I guess you've removed and cleaned the jaws in the chuck.
The condition you are describing, 15 thou in 4" is pretty extreme. First, remove the chuck and check the squareness of the spindle nose as follows.
1, Lightly stone the mounting face of the spindle nose to identify and remove any burrs. No high spots should show up.
2, With a mag base on the cross slide, and dial test indicator tip on the face of the spindle nose, rotate the spindle and check the face for runout. Anything other than zero (or darn near zero) would be surprising, and bad. A machine problem, not the chuck.
3. With the indicator still mounted to the cross slide, adjust the location of the indicator tip so that you can traverse the the spindle mounting face from front to back, covering as much of the spindle face diameter as possible. Basically tracing a chord across the spindle face. Again, anything other than zero would be surprising and again mean a machine problem. Without moving the carriage or indicator, rotate the spindle at 60 degree intervals and traverse the cross slide front to back at each location. All readings should be near zero. Anything else is a machine problem.
If the spindle nose checks out ok, and I hope it does, next is the chuck.
4. Lightly stone the mounting face of the chuck to identify and remove any burrs. Load your chuck into the lathe. Load a known straight rod at least 1" in dia x 8" long into the chuck, with 5" sticking out. Set the mag base on the cross slide and the dial indicator tip at 12 o'clock on the rod at one inch away from the chuck jaws. Rotate the spindle to find the high point of the rod. Note the TIR and mark the 12 o'clock high point of the rod with a sharpie. Also mark the chuck, the nearest jaw, and a visible part of the spindle nose, all at 12 o'clock. With the indicator still at the 12 o'clock high point on the rod, move the carriage along the length of the rod. You said that showed a 10-15 thou rise on the indicator, correct? Rotate the spindle to verify that the high point at the end of the rod is aligned with the high point near the chuck. Mark the rod at 12 o'clock.
You should now have a series of lines marking the high point at 12 o'clock on the rod, chuck, jaws, and spindle. Also, at this time, number the individual chuck jaws and chuck slots as 1,2, and 3.
5. With the rod and spindle still in the 12 o'clock orientation, move the indicator and its base so that the indicator tip contacts the front of the rod, near the chuck, at 9 o'clock. Traverse the carriage along the length of the rod and hope to see zero movement. If you see more than zero, adjust the indicator tip so that it follows along the centerline of the rod, not too high, and not too low. If you are traversing the length along the CL of the rod, the curvature of the rod would only show a 2 tenths movement of the indicator. But if you are 50 thou above centerline, you'd see almost 2 thou movement. Kind of a false positive. The point of this part is to find out which way things are "bent", and get some baseline marks in place. So, everything should be marked, and "bent" upward toward 12 o'clock.
Up to this point, there should have been no surprises and everything should have gone as described. I hope, LOL.
6. OK, now to finding the real problem.
With everything back at 12 o'clock, re-set the indicator to touch the top of the rod at it's 12 o'clock high point near the chuck face. Rotate the spindle and set the indicator to read zero at 3 and 9 o'clock and high and low at 12 and 6.
7. Check the rod. Leaving the spindle at 12, barely unchuck the rod and rotate it 180 degrees to 6 oclock. Indicate your rod again at 12 oclock at both ends. The indicator readings should not have changed. If they changed, either the rod is bent or the chuck repeatability is really bad. But lets assume the readings stayed the same.
8. Leaving the rod chucked in the chuck, remove the chuck and reinstall it rotated 120 degrees. Now, with the indicator again at 12 oclock and the chuck and rod at 12, and the Spindle off by 120 degrees, recheck the dial indicator readings. If the indicator readings are the same as before, that means the runout is following the chuck and not the spindle. Move the chuck to the next set of holes to verify that the runout stays with the chuck.
So what happened? Did the runout follow the chuck or the spindle?
Move everything, including the chuck and spindle, back to the 12 o'clock position.
9. Now check the chuck jaws. With everything back at 12, verify that the indicator readings are the same as they were in step 6.
10. Remove the bar and the chuck jaws after making sure that everything is still marked. Reinstall the jaws indexed to the next set of slots, and rechuck the bar. Align all the marks at 12 oclock except that the chuck jaws will now be off 120 degrees. Set the indicator back up and check the top of the rod and see if the readings are the same as in step 6. My guess is that they will Not be the same. You'll probably find that the runout has kind of followed the jaws, but not exactly.
11. Now, rotate the spindle to get the high point of the bar again at 12 oclock. So, is the high point still in relation to the original jaw? If it followed the jaw, you are lucky, because it might just be a jaw problem.
12. You will probably find that the readings are best with the jaws installed in one particular position, and that the runout is tied to the location of the jaws. As previously suggested, try some different diameter bars to see if its a scroll problem, too. But if its mostly the jaws you can clamp a ring deep in the chuck and bore them out just enough to clean up. Make sure to permanently mark the number one jaw position. After boring the jaws, you'll have to hand finish the section of each jaw that clamped onto the ring.
I had to go thru this procedure with my 5" 3 jaw and I got the runout an repeatability from about 9 thou down to 2 thou. I would have liked better, but that is why we have 4 jaw chucks, right?
My apologies for the long winded post, but i hope it might help somebody.
Lloyd-ss