Last week at the Supreme Court, a lawyer made what seemed like an unremarkable point about registered sex offenders.

“This court has recognized that they have a high rate of recidivism and are very likely to do this again,” said the lawyer, Robert C. Montgomery, who was defending a North Carolina statute that bars sex offenders from using Facebook, Twitter and other social media services.

The Supreme Court has indeed said the risk that sex offenders will commit new crimes is “frightening and high.” That phrase, in a 2003 decision upholding Alaska’s sex offender registration law, has been exceptionally influential. It has appeared in more than 100 lower-court opinions, and it has helped justify laws that effectively banish registered sex offenders from many aspects of everyday life.

But there is vanishingly little evidence for the Supreme Court’s assertion that convicted sex offenders commit new offenses at very high rates. The story behind the notion, it turns out, starts with a throwaway line in a glossy magazine.