My first JABC hammer gun had the fake Damascus. I don't remember what it said on the rib but it was probably something like London Fine Twist. It was obviously a fake because the "twist" pattern continued from one barrel, across the under rib and on to the other barrel. Ditto 2-P's statement that you shouldn't expect them to be better, or even as strong, as a set of well made Damascus barrels. I have seen some marked "Damascus Finish" as he said but the one I had gave no such indication of fakery. A likely explanation on the Parker was that the sleever simply used the original Parker rib. Look closely at the under rib on your gun, Mike. See if the pattern runs right across it, like mine did.
