That´s all why I addressed the need to know the real material.
Loads: the loads our proofhouse use (and used as far as known 200 years back) are the same for all BP proofs. In a short barrel, you usualy don´t even get the highest peak of pressure the load can create, due to the fact that the monstrous powder column clears the muzzle before even getting fully ignited. And once it can be ignited, then the barrel must withstand it. The object is, that because with BP there´s almost impossible to reach pressures over about 25 000-28 000 PSI* (depends on grain and moisture content) with any minorly sane load-just due to burn properties of BP, to expose the barrel to the highest pressure it might be exposed to during use. Doublecharging, doubleballing can be easy-shooters are dividen just into two groups: those who did and those who will. So the integrity of the barrel must not get any damage or stress, which might compromise the safety and, if possible, also any other aspect of the gun. So triple charging, double or triple balling (in weight, not actual pieces of lead) is a good way to go to be sure.
It can be applied for a reduced load proof- which must be therefore visibly specified on the barrel. But even historically, as to my knowledge-a low use, low quality smokepole were tested for 2,5-3 charges, double balling, serious guns triple charge, triple balling. The too frequently cited doublecharge, doubleballing being absolute minimum, or for a gun which is otherwise brought onto edge of limits of the recognized best quality material and manufacture. (Like these shotguns, which have the barrel contour derived from the pressure curve for max load, being just a tad oversized, with 0,9 mm of steel in the choke for 12x3" ga @ 34" barrels-nothing personal, but just due to material it´s not equal case here. Also, if I understand, you want to try your first barrel-and pushing the limits on your first-I think we understand each other here).
*Counts only for 3F and coarser. 4F and other very fine grains are other beast, in which there´s realy not known the mechanism and chemism they burn, esp. under high pressures-which is definitely somehow different from the burn processes of coarser grain. But it´s recognized that it can do (sometimes, but not known why) worse pressure peak and damage than a lot of smokeless kinds.
As to material:
that´s right the problem that the strenght of what I have seen comonly called "mild steel", varied from 320 to 840 MPa of tensile strenght and 53-85% of that for yield strenght, with very great differences regarding shock load properties. While 90+% of the materials are way under 480 MPa tensile strenght. The 12L14 is a low carbon, low sulphur, low lead automation steel, which has min. tensile strenght of 540 MPa and min. yield strenght of 415 MPa. And has the capability to stand some shocks just up to yield tension with no to minimal change in structure or measuremnts of the part. 8.8 class nuts up to M12 (M14 maybe also) can be made from this steel, if I recall correctly.
The Jaeger barrel you describe-I suppose the 5,65 mm is wall thickness just at the breech shoulder and accordingly I fed the calc for it. So-if the plug is installed with enought pre-tension, that it would not set-back momentarily upon firing, thus not letting the gasses into the thread (so is properly seated), then using our proofhouse load is just about right on spot to create the yield strenght equal tension, just a tad under the point where permanent deformation occurs. So, if the barrel is OK, then it´s OK throughout all 3 proof rounds (due to the steel properties), and so it will withstand the service loads. But if something is minorly wrong, it would fail the proof.
So your Jaeger barrel is spot on to pass the abovementioned tests with flying colors.
The statement about 12-14 mm for 62 cal. counted for safety for 1000´s of rounds with even such steels as 1000 or 1005, steels used for deep draw etc. A kind of grabing some piece of metal on the drawn-can factory scrapyard (absurd for sane person, I know).
Of course, if you need just one or five shots of just the quality of having the desired MV, you can use water pipes and soutable load-it will hold these few rounds, albeit being stressed on lmost 100% of the limit. But no consistency, no longevity...
DOM hydraulic tubing is anything, but as to my knowledge (maybe just still a language incapacity) DEFINITELY not a "mild steel". They are of carbon or low-alloy steels, with tensile strenght between (900)1050-1430(1570) MPa, with yield strenght 82-88% of TS-as I derived from different manufacturer´s specification for different diam.´s and wall thicknesses. It´s @!*% well equal to 4130 heat treated CrMo barrel steel, even higher in some cases.
BUT:
-part of the strenght is obtained from the mfg process-rolling and drawing
-stability of the material, as well as of the dimensions, is maintnaited by ballance of the stresses involved and induced into the material during the mfg process, which are only in some cases partially released by heat treatment
So-if you cut off some layers of the material during turning the barrel, it may seriously compromise the strenght of the tubing. Totaly impossible to calculate how much.
If the barrel is made from stock for making hydraulic cylinders, you have good chance that you´re OK, because it´s usualy heat treated in such a manner to allow welding and turning without warpages, as well as not making fruther heat treating mandatory. TS seems to be in (750)850-1300 MPa range, YS in 73-90% of TS. This stock is usualy supplied in a form of thick-walled hydraulic DOM tubing, so for uninformed people it´s the same.
Conclusion:
IF the blank you turned is from DOM tubing, scrap it or proof it for tripled service charge (about 3x 50 grains), 2 balls. And watch with micrometer for any 0,0001 inch of difference in dia, esp. around the breech face. I would advice look for elongation too. If OK, engrave the max. service load on it, just to be sure it won´t kill someone when it´s not in your hands. It´s not PC/HC, but the material is realy deeply different from anything a man would expect in a tapered barrel.
If the blank is from the hydraulic cylinder stock, I would proof it for triple charge (3x60 grs), triple ball. In the worst case (lowest TS and lowest YS/TS ratio, you´re with the YS somewhere in the ballpark ot TS of 12L14, so the 4 mm may do the job well. From my point of view, the biggest problem you´re facing is perfection of the breeching job-clean run-out of the thread, square shoulder with no excessive cut into the wall, proper close thread tolerance, proper pre-tension of the breechplug face to the shoulder. There´s no big margin, so it can kill the barrel.
I hope I didn´t forgot something...
Also I think I should add some disclaimer about the fact, that although I stand behind my calcs, the entry data esp. for the tubing and cylinder stock may vary due to mfg and material. Check for the piece of metal in hands is needed.