Is it permissible for a student to record lectures without the professor's permission if he cannot understand or remember them except by recording, despite having paid money for understanding, and with the student's pledge not to leak the recording?

You are not permitted to record the lecture without the lecturer's permission, even if you do not intend to publicize it. Therefore, you should either convince him to give you permission, or do not record without his consent. You can engage with others who do allow recording.