Scope - Torquing and Lapping Rings for Accuracy...

AZ
Yours is a "Counsel for Perfection" - nothing wrong with that at all ... but :)-) ) Show me a perfectly straight barrel ('BNIB' or unused) - it wont be perfectly straight, then show me that same barrel on a very cold day after 3-4 thousand rounds have been thro' it + a bit of general owner knockabout !


I cannot speak for 'Kris' here but I think we are on the same script... approx.

Here's my methodology -
MOUNTS
I choose good mounts/rings - I distrust those that clamp from one side (& also have 2 diff. sized clamp options). A pretty good test of any rings is to run your finger around the inner surface of the rings when they are tightened ('hand tight') off the gun. Many times you can feel a ridge between the upper & lower sections (& the Mk1 Human Finger is capable of extremely fine judgements, folks!), this tells you they are not matched or machined from one piece, then matched - you could try turning the top section or swapping with the other set, most likely it wont improve matters. SO, what to do?
Buy a set of good quality mounts in the first place - BKL get my vote but there are others (some PB rings clamp vertically).

The reason for all this is have the best possible vertical alignment of Barrel, Rifle and Scope

SCOPE
I favour a brand of mid-market Euro scopes, before they go on a rifle I optically centre them (using the mirror technique or sometimes the click method).
Using a bubble level and a plumbline I do my best to have the scope sitting correctly in the mount cradles before tightening down (diagonal bolts etc).
Now mounted into the rings I use an optical boresighter (largely out of interest, they are not essential) - I fully expect to see small variations which equate to around 1-3 "clicks" in both Windage & Elevation from scope to scope. I also do a 'dry run' Box Test using the boresighter, if OK, I then properly establish the Zero by actual firing (remember that !) When the usual further adjustments are made to bring the Reticle to the Point of Impact I reset the Turrets or make a note of the settings.
Then I SHOOT the Box Test.

If all OK - I'm done & dusted !!!
(at least for a short while, monthly Zero checks are advisable).

In my view a lot can be put down to "BELIEF" in your setup, if you lack confidence in 1 area (of equipment) then ...
 
"kris"good info on Torquing and Lapping rings.however there are more important things like that.

I have also found, that first of all you need to optically center your scope, I spin it on the gun with the top of mounts missing. Spinning it at high mag focused to a distance target adjusting/minimizing the arcs the middle of the crosshair is describing, other people do it in v grove box or using a bathroom mirror whatever works for you.

After centering you must understand that you CAN NOT touch the turrets AT ALL to align the crosshairs with the barrel /POI. you have to find other ways of doing it.
you may have to rotate or swap around two piece mounts, shim horizontally or sideways if needed, use adjustable mounts you name it.

If you disagree:
try to shoot a 8yard very small target. if your 100yards is perfect and 50yard is perfect you may be very surpriced by how much you may be off on close targets. I am not talking about vertical holdover holdunder, but the horizontal axis!
Impossible? just try it if you think your scope is perfect. 
it does not make sense -does it.

heeee heeee

if you don't listen you can be spending big time on great named scopes and never be happy with them.
Why do they bother to put windage and elevation adjustments on scopes then? Why not just make them centered and non-adjustable? Force you to use adjustable rings?
 
BigTinBoat - the situation you imagine there (no internal adj.) actually applied for approx 100 years !
Some very fine quality sniper scopes relied entirely upon external adjustable mounts - please note ADJUSTABLE :).

I'm afraid AZ exists in another world to me (anyway), as I said before I EXPECT to make regular 'corrections' to my rifle scopes over time and rely upon the repeatability factor even though I mainly use First Focal Plane scopes.