Yeah, those carbon remover cloths are a wonder! They used to sell them in drug stores as miracle cloth or some such. Pro-Shot Products sells them, if you can't find them locally.
First use a good gun cleaning solvent. Let it sit for a while and then scrub with a piece of rag and a bristle brush. If that doesn't do it to your satisfaction on a SS gun use one of these.
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Remington also markets a cloth for stainless guns like the Klean Bore Lead Away. Still takes a little elbow grease, but they sure work good. Been using them for years. I used to have some patches of the same material in a little container like a 35mm film canister. They are long gone. I cut a couple of small strips for cleaning above the forcing cone.
I also use the Lead Away cloths. I cut a piece 2 or 3 inches square and wrap it around a large wood "tongue depresser" thingy bought at a hobby store. Works great for scrubbing the front of the cylinder.
A bit of Flitz or similar polish -- even Brasso or silver polish -- on a rag, with a smidge of elbow grease, will take it right off. That's basically all a Lead-Away cloth is.
I use wire brush and Break Free CLP... Its gets a good amount, other than those deep carbon rings... Mostly clean is good enough becuase they come back...
Birchwood Casey makes a great lead remover & polishing cloth. It is meant for Stainless guns. It easily removes the cylinder burn rings and it does an excellent job of polishing the entire gun.. It's the best product I've seen for removing the stubborn carbon deposits.
"Lead Removal Cloth
Lead deposits on the face of revolver cylinders and similar places can be removed with a lead wiping cloth prepared as follows.
Mix the following ingredients
500 gr - 400 grit or finer aluminum oxide powder
450 gr - kerosene or #2 fuel oil
4 gr - lemon oil (for a more pleasant smell)
5 gr - ammonium chloride
Evenly saturate a soft thick cotton cloth or flannel with the solution and allow to dry. (There is no reason it won't work wet though.)
Carefully remove any very heavy lead deposits with a scraper and then wipe the remainder with the cloth to remove."
I put a coat of past wax on the cylinder before I shoot and buff it off. It leaves enough of a barrier so the black grud does not stick so well. Wipes right off.