"Knowledge of how to create an atomic bomb" would still qualify, but also gets torn down pretty quickly - that information isn't owned, and can't be restriced to trustess - the "information" simply derives from science, and is non-excludagle to any intelligent society no matter what you do to try and "exclude" them. So even in that case, you must fall back to regulating the physical - limiting supply of nuclear materials, policing/spying on use of facilities that 'could' be used to create them, or so on.
Information freedom isn't just some 'inevitable truth' - it's the nature of information itself. Arguing against it isn't even futile, it demonstrates a fundamental lack of understanding. (Which I suppose could have been guessed at by the spelling of the first 3 words of your comment, but I suppose others can get the benefit)
Actually, I am pretty sure we don't. Those codes come in pairs and are all subject to two person control. Also, those guys that turn the keys can use their own judgment to veto the launch. In the event a leak was known or an unknown reason to launch came through I suspect they wouldn't.
Those codes won't let them re-target the missiles either, we will be firing them at whatever the predesignated target is. I am not sure if anyone else is up to the task of firing back in the ten or so minutes before all of our cold war enemies are erased.
So how about back to the practical example about what information can go free in the event of government transparency. Surely there are some plausible examples that can also make your point.