4 thou RO I would not be happy with but it will most likely not be the cause of your accuracy issues, as pointed out back a bit.
I first became very anal about runout when I used to shoot Fullbore.
It seemed to be a random cartridge to cartridge thing as I loaded them (yes I partially seat the projectiles and the turn 180 degrees and complete, always have).
(When I first started handloading, again for fullbore, I rolled the cartridges across the top of my loading desk. If they didn't visibly go madly up and down at the tips they were deemed vg.)
So in a batch of say 30 loaded I would have maybe 6 that I deemed to have excessive runout. (And some of these had greater than 4 thou runout.)
These I would put aside for the easier ranges eg 500 yds.
Eventually I would end up with so many that you have to either strip them and reprocess them or fire them.
I then used to fire them on club days at whatever range was being shot.
And....my possibles still kept coming in and without reduced centre count.
I'm not trying to say runout control is not important, it is. A projectile starting of on the piss into the bore never corrects itself.
It's just that runout is part of the whole package. I would rate neck tension as important if not more so than minor runout.
To get to the bottom of runout you need to examine your cases at every stage of the process to find where the issue is being induced.
Using your runout gauge measure the necks as they come out fired from the rifle. You should see maybe 1/2 to zero (preferable) thou runout.
Check at each stage of your reloading ie after neck sizing or FLSing (which ever you do) and so on.
Final check before seating. If possible run a gauge inside the necks as well as outside. If you detect runout at this point I have found no seating die will correct it.
(I have Redding Comp, Wilson inline, Forster Micrometer BR seaters.)
I haven't mentioned the basics here like good neck chamfering etc.
The biggest causes I have found to be Redding Body dies (necks totally unsupported), some bushes in bushing dies, just plain poor quality dies in general.
The humble much maligned, by some, Lee Collet die I have found to be very good at keeping necks straight. Again straight necks going into a good seating die will usually give straight cartridges out.
I have completely changed my loading process. I now anneal after every firing, FLS (Forster die with expander removed, 0.5 to 1 thou shoulder bump), then expand necks using expander mandrels to generally give 2 thou neck tension.
I get zero to 1 thou runout measured at the projectile tip. I could never consistently achieve this with bushing dies.
As pointed out by winaa, quality seating stems are important. My seating stems are honed to give good contact on the projectiles. Unfortunately it means I have ended up with multiple seating stems. VLD stems for VLD projectiles etc.
Now, having said all that above, I just load my 223 with standard dies and it shoots tiny wee groups.
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