Ok, so the principle of breaking in is that the first several rounds down the barrel polish the surface of the bore and make it smooth so it fouls more evenly for the rest of its iife.
The benefits are supposed to be accuracy and reduced fouling so the rifle can fire more shots between cleaning and the accurate life of the barrel is longer overall.
The problem is that barrels vary in their potential accuracy so doing a controlled trial to demonstrate benefit or none, compared to starting straight into shooting long uncleaned strings is extraordinarily difficult and the amount of benefit is said to vary with calibre, bullet type, barrel type .... So it is unlikely anyone will produce experimental evidence to prove the value of breaking in, even if it were highly beneficial. We therefore rely on recognised principles of engineering and advice from experts such as barrel manufacturers. You can google and find these and the preponderance of highly successful competitive shooters do breaking in.
There are two main sorts of fouling: powder/carbon fouling and copper fouling. The theory is that if these build up on the new barrel surface, some patches get polished nicely while others accumulate successive layers of carbon and copper which peel off at random leaving a chaotic surface which changes with each shot. Carbon is relatively easy to clean off with a powder solvent but you really need a bronze brush to get it all off. Copper needs special solvents and can be very time consuming and the solvents are expensive. Using a good one piece cleaning rod and a bore guide will enable you to clean without damaging your rifle.
A common strategy is to fire 5 to 10 shots cleaning in between each one, then gradually increase the number of shots till you are shooting strings of 5. Once the bore is considered to be broken in, most people will clean it for carbon after each day's shooting and perhaps every 50 to hundred rounds for copper (this varies a lot between shooters !). During the break in brocess you can be sighting in your rifle and slowly checking grouping capability and trajectory and even go out for a hunt or two, so its not wasted ammo, nor barrel life.
Barrels typically last 2000 to 5000 rounds so if you do a lot of shooting and think you will wear your barrel out and replace it one day (as many good shooters do) then you could consider the break in process an additional profit for the manufacturers although generally the ammunition and cleaning gear, not to mention your time and the main costs.
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