Question: My understanding is that the Catholic Church teaches that when the Pope speaks ex cathedra, he is infallible, because what he says in such instances is divinely protected from error. There are a lot of questions that the Church considers open, e.g. the question of whether infants who die without being baptized are saved. It seems like it would be a great idea for the Pope to *attempt* to assert ex cathedra that they are saved. Either the pronouncement would succeed, in which case we would know with certainty that it is true, or else something would prevent the pronouncement from going through, in which case we would have a hint that it is false.

I assume that I am misunderstanding something about papal infallibility and there is a catch somewhere. **What's the catch?** Naturally, I'm interested in the teaching of the Catholic Church.

Here are a couple of possible answers:

* Technically, the Pope *could* "exploit" infallibility in this way, but it would be immoral.
* If the Pope decided to try this sort of thing, he would be divinely prevented from making the ex cathedra pronouncement regardless of whether the doctrine in question was true or false, so we wouldn't actually learn anything from the exercise.

In either case, I'm left wondering: What exactly differentiates "legitimate" ex cathedra pronouncements, like the pronouncement of the immaculate conception in Ineffabilis Deus, from the sort of "illegitimate" ex cathedra pronouncement that I suggested?

Answer: 
You massively misunderstand the process of promulgating doctrine. It's not a magic process where God causes true statements to be generated out of thin air. Doctrine is developed through prayer, research, consultation, discernment and much more, and involves the whole church, not just the Pope. The belief is that God guides this process to ensure it is correct. Shortcutting the means would be equivalent to cheating God.
"it would be a great idea for the Pope to attempt to assert ex cathedra that they are saved"

Probably not. God would probably have to kill or otherwise divinely stop the mouth (or pen in this case) of said pope before he did so, as stated by another person here, else the Church would have been overcome by Satan (the true faith is anathematized and lost and error taught in its place) in which impossibility the infallibility of the Church (in an extraordinatary sense with regard to the successor of St. Peter—due to the fact that he overrides the authority of all bishops and has the authority to bind all Catholics to a certain interpretation of the deposit of Faith whereas individual bishops do not—and in an ordinary overall sense with regard to the world's bishops) exists.

Papal infallibility precludes that such a 'trial dogma definition' could ever happen, ever. It would breach the infallibility of the Pope (more a descriptive fact about the Church than a 'power').

That is, your question is like 'what if water wasn't H2O' and is meaningless to ask (sorry, in the logical sense, not that it has no value in being asked).

Papal infallibility is a necessary consequence of 1) the infallibility of the Church as a whole (that the gates of Hell will not prevail against it—which a new faith/false doctrine being taught would constitute) and 2) that the Pope has the authority to teach and bind all to his teaching, in a way other bishops cannot and therefore do not require, and are not protected by, this kind of infallibility. They possess an infallibility which basically means no amount of bishops teaching heresy x will result in the same overcoming of the Church by Hell—a much more descriptive 'infallibility' than even the Pope.

It isn't some magic power.