Well, I have some experience on this. My car was broken into last year and my backpack with a Ruger 9mm autoloader was stolen. I reported it to the police. However I was unable to find the serial number for that gun at home. So I just wrote it off. Then last month I got a letter from the USDoJ stating that it had been found in a drug raid not far from where it was stolen and they were seeking forfeiture. Apparently they have an illegal gun database now. As I’ve suspected for years. The gun has never been registered or otherwise recorded in any fashion except by the instant check and form filed at the time of purchase.
That form (4473) that you filled out at purchase traces the gun to you as the original buyer, assuming it was a new gun at the time. When guns are recovered by any form of police a check is started. It goes from the mfr to the distributor to the store that sold it. The store looks up who they sold it to and you get a call.
Note for future reference: if you lose a gun, file a police report even if you can’t describe it completely. Note also that, in your case, Ruger has always serialized their guns so that if you buy one new, it will always trace to you. FWIW that whole tracing operation rarely takes more than a day. Some states have a law requiring reporting missing guns. Failing that can get you a nasty phone call.