@Alex Wheeler shows exactly what I mean.
Restating with different wording so you can correct me if I have it wrong.
As I understand it,
@Curious said, make sure the chamber/throat/breach face are all coaxial to the bore at the breach. The muzzle doesn't matter at that stage.
However, if you indicate the curvature of the muzzle exit, what is in between does not matter. Knowing the "crown" of the bore will let us make the breach coaxial to the action/chamber/throat. If we mark the point at which the muzzle is at it's highest deviation from the breach axis and tune the barrel/action interface so that this deviation is centered up then there will be less left right correction.
I'm going to add one thing to the above paragraph. An observation. Where I know the direction the muzzle "crown" as in "crown" like a piece of lumber like
@Alex Wheeler s picture. Almost always, that is the direction the point of impact is going to move with barrel heating. Example: My new DTA, I know the muzzle points "up" from being completely axial with the bore at the breach. When I get to round 6 of say a 10 round string, the point of impact starts to move up. The hotter the barrel gets the more the point of impact moves. This is not the only rifle I have where I have POI shift with barrel heating.
So this brings me to
@butchlambert s comment
If the "crown" curves up and the POI moves up with barrel heating, this can be easily mapped. If it is at a random angle mapping this POI shift and factoring in wind effects make for a much more difficult shooting solution.
So what I plan on doing is adding an additional step to setting up a barrel in my lathe. Indicate the breach axially true to the axis of the head stock. Determine the point of deviation at the muzzle by measuring the high and low spots with another indicator rod. Touch up the tenon so the action locks up under torque with high end of the barrel curvature pointing up relative with the action center line. After that, when crowning and turning for muzzle devices such as brakes. True the muzzle to the axis of the head stock to make sure the muzzle crown shape and threading for barrel devices is axial to the bore of the muzzle end.
Hopefully, like my DTA, when the barrel heats up during fast or long firing strings, I will know how much the barrel will rise.
Objective data: My DTA 300WM barrel had a loose barrel extension. Just hand tight. Not an issue for safety. However, during shooting strings the POI would move "unpredictable". Last Wednesday I torqued the barrel down to the extension. Today POI shift during long strings was straight vertical.