on/off the/somebody's books
off the books
Off the official payroll; paid in cash, without any official employment paperwork. I worked off the books in my uncle's restaurant for a few months during the summer. It's no secret that many migrant workers are kept off the books at major agricultural and construction operations around the country.
on the books
Set down in writing or an audio or video recording. Today is officially the hottest day on the books. His statement is now on the books, so he can't dispute that he said it.