May I suggest that you read, if available, the section relating to paper vs brass case shotguns in The Modern Sportsman's Gun & Rifle by J H Walsh (Stonehenge)...the author goes to some length describing the dilemma of bore - gauge classification at that period.
A 12 G gun which is chambered to take the standard 12 G cartridge but has the barrel bored on the narrow side would have been stamped as 13, a narrow bored 16G would be stamped as 17 and so on. Basically the gun was stamped with the barrel diameter as per the definition of bore size for muzzle loaders. As you know the cartridges for each gauge were identified with a headstamp signifying the chamber size of the gun. After the "chamberless" gun for brass cases was introduced there arose considerable confusion as to which cartridge type was appropriate to use in a gun which may only be stamped as , for example,"12". Latterly the cartridges had head stamps to clarify the problem - a chamberless 12 which accepts a 12G cartridge but with a barrel bore diameter matching that of 10G- those shells would be head stamped as 12 / 10.
Back to the issue in hand:
a muzzle loading 4 bore will have a true 4 gauge barrel, based on 4 lead balls of that calibre weigh 16 ounces.
a thin brass case 4 bore also has a true 4 gauge barrel , give or take a few thou.
a paper case 4 bore generally would have a 6 bore barrel although labelled with the misnomer of 4 - ie paper and brass case guns will have similar chamber dimension but different barrel diameter. The ballistics given by paper case shells in a chamberless/brass case gun were notably weak, the use of brass case shells with their heavy load of powder and shot in paper case guns was not recommended! ....until gunmakers started to manufacture (wildfowl) guns bored specifically to handle both types of cartridge...progress of sorts?
