The way it works here Florida is that the buyer completes the 4473. The FFL then reviews the form to determine if the the buyer is not prohibited from purchasing a firearm. Then the FFL makes a call-in background check with the Florida Department of Law Enforcement(FDLE). If the buyer is denied then the FFL can not transfer the firearm. After 30 days (the appeal time) the local law enforcement agency who has jurisdiction where the purchase was attempted is notified by the FDLE and they pick up the form 4473 and investigate the buyer for lying on the form which is a federal and state felony.
That's IF the FDLE can find the applicant. If the applicant lied on the 4473 about eligibility to buy a gun, he/she probably also lied about their home address. I guess an applicant would leave fingerprints on the form, but would fingerprints on paper remain detectable and readable after 30 days? It seems to me that the skin oils that leave the prints would have dispersed through the paper's fibers and become so blurred by the that time that they would be almost useless.