What would happen if a manifest heretic were elected as Roman Pontiff, and he began to reign over the whole Church?
The answer is found in the history of the Church. Pope Silverius reigned in the 6th century as a valid and faithful Roman Pontiff. Vigilius desired to be Pope, so he promised the Roman emperor and empress that he would support their heresies and install a heretic in the See of Constantinople if they made him pope — which they did. But since Silverius was still the valid Pope, Vigilius was an antipope. And he promoted the heresies of the emperor and empress.
Then Silverius died and Vigilius was installed at Rome as the Roman Pontiff. He was accepted by the clergy and people of Rome, and by the body of Bishops in the world.
Bellarmine: “From this time neither error nor feigning of error was discovered in Vigilius, but rather, supreme constancy in the faith even to death, as it shall appear. For he received with the pontificate the strength of faith and he was changed from a weak chaff into the most solid rock.”
When a heretic is elected Pope, the prevenient grace of God, which no one can resist, immediately vanquishes all apostasy, heresy, schism, idolatry, sacrilege, and blasphemy from his person, and gives him the gift of truth and of a never failing faith.
No valid Roman Pontiff has ever taught grave error, nor failed in faith. No valid Pope has ever been an apostate or heretic or idolater.
RLCJ
the reproach of Christ 
«No valid Roman Pontiff has ever taught grave error»
I don’t understand how you can conform this statement to the historical case of John XXII.
Do you think his error was not “grave”? Or do you think that his statements were not “teaching”?
I agree that obviously no Pope can fall into heresy. John XXII was not heretic (the dogma of beatific vision was formally pronounced by his successor). But I don’t see how deny that he taught a grave error.
John 22 clearly stated that his view was opinion, not a teaching. He allowed disagreement. He stated his intention to work toward a magisterial decision. He changed his opinion before he passed away. His successor made the magisterial decision and issued the dogma. Theological opinions are not grave teaching errors. And it was not a grave error of opinion, as it does no harm to souls to propose an opinion, and encourage theological opinions to the contrary. Also, an error on when we receive the Beatific Vision does not harm souls (though now it is dogma).
Interesting. Can you recommend a website where I can read some quote of the Pope’statements? English or Latin are ok. Thanks very much.
“John, at that time, really thought that souls would not see God unless it were after the resurrection: others so reckoned when still it was lawful without danger of heresy, since still no definition of the Church had gone before him. John, moreover, wished to define the question, but while still preparing and in consultations, died, as Benedict XII, his successor, witnessed in Extravagantes which begins; Benedictus Deus, the whole of which Alphonsus a Castro relates.
“Furthermore, John Villanus relates that Pope John, before his death, partly declared and even partly recanted his opinion. First, it is on good evidence that he never had it in his mind, although he had spoken on this matter, to define the question, rather only to treat to discover the truth. Next, he added that John already thought the opinion was the more probable, that asserts the souls of the blessed enjoy the divine vision even before the day of judgment, and he embraced this opinion, unless at some time the Church would have defined otherwise, and he subjected all his teachings freely to its definition. This retraction simply teaches that the mind of Pope John XXII was always good and Catholic.”
Bellarmine, Robert. Papal Error?: A Defense of Popes said to have Erred in Fatih (pp. 104-105). Mediatrix Press. Kindle Edition.