If he tries to speak ex cathedra will the chair zot him right out of it? That sounds like a joke but there’s a serious question underlying. A “bad” Pope, is he supernaturally enjoined from making an infallible statement in the understanding of Roman Catholics? It’s all so mechanical and ritualistic, it’s just alien to me, I’ve got to admit.
First of all, “ex cathedra” does not mean actually, physically sitting in some particular chair.
There have been two exercises of “papal infallibility”—i.e., actual statements by the Pope which explicitly affirmed that the Pope intended to teach the entire Church infallibly, in 1854 (Immaculate Conception) and 1950 (Assumption), and 1994 (ordination of females).
In all cases, the statements were the result of long study by scholars of the historical record, and inquiries sent to every bishop in the world, specifically on the question: Has this dogma been a part of the Deposit of Faith from the beginning?
Worst case possible regarding the Synod is that Francis emits a statement as vapid, juvenile, and destructive as the ones Kasper, Wuerl, O’Malley, et al., have been putting out.
If he does that, he WILL NOT assert that he is teaching infallibly.
At the risk of sounding superstitious, I will say that IF Francis were stupid and wicked enough to make up his mind to claim to teach infallibly something different from what the Church has always taught about marriage, Communion, etc., that he would die the night before.
Seriously, if he DID do that, he would cause a schism. The real Catholics would reject this new teaching, and elect a new Pope.