sign off on (something)
To give express, formal approval of something, often by literally signing a document. We're just waiting for the boss to sign off on the project before we start working. My thesis supervisor has to sign off on each chapter I write to make sure I'm staying focused on the topic.
sign off on something
to sign a paper, indicating that one has finished with something or agrees with the state of something. The publisher signed off on the book and sent it to be printed. I refuse to sign off on this project until it is done correctly.