Is the movement of the lips considered a secret in an oath that makes the oath binding?

One of the conditions for an oath to be binding is that the swearer must utter their oath by moving their tongue; otherwise, the oath is not binding. This is based on the Prophet's (peace be upon him) saying: "Indeed, Allah has pardoned my Ummah for whatever occurs to their inner thoughts, as long as they do not act upon it or speak of it." Moving the lips is stronger than moving the tongue, based on the Prophet's (peace be upon him) saying: "I am with My servant when he remembers Me and his lips move with My remembrance." Based on this, the oath you swore is binding and obligatory, and you are liable for expiation if you break it.