Yeti, you seem to be looking for a Third Secret that will tell you "literally" what will happen in the "end times."
Each end times prophecy, Biblical or private revelation, will have some elements that are literal, mixed up with elements that are figurative. The challenge is to determine which is which. It is not a perfect science and depends more on prayer and humility than science anyway.
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Oh, I think I see the problem here. I don't know of any indication that the third secret is a prophecy of the end times. I'm not aware that Our Lady or any of the seers said this. Why do you think it is?
Do you deny that there will be a supernatural intervention that separates our current era from the new era? As St. Augustine explains, in our current era, the Church is a mixed society (containing both true Christians and bad). But in the coming era, the "era of peace," the Church will only contain the good Christians and these Christians will be "immortal." That is not a description of our current world. In our current world, all men are mortal. Therefore, the new era will be something completely different from what we now know. Do you deny this?
The Faith teaches that the world will last until Christ comes again at the end of the world to judge the living and the dead. That will be the last judgment. After that, every human being will be either in heaven or in hell. The Church Militant and Church Suffering will no longer exist; only the Church Triumphant. That state will last for all eternity. Most of this is Catholic dogma. The statement of St. Augustine is referring to the state of eternity after the end of the world and the last judgment. The supernatural intervention that you mention is the second coming of Christ, the end of the world, the general resurrection, and the last judgment.
I'm not sure what you're describing here, and it doesn't seem to match this all that well.
If you don't deny that the new era will be something completely different, then why do you worry about the Church's geographical headquarters? It is irrelevant.
Rome will not be the headquarters of the Church after the end of the world because the Church Militant will not exist any longer, and probably Rome itself will not exist either. My concern is that it seems to violate the canons of Vatican I, Trent, and other councils to say that Rome will cease to be the head of the Church
before the end of the world. It is defined that the
Roman pontiff is the successor of St. Peter and the head of the Church. Theologians have said this must mean that the holy see cannot be moved to any other diocese apart from Rome, since then the bishop of that other diocese, and not the Roman pontiff, would be the head of the Church.
But the text of this third secret says that the authority of the Church will be moved to Fatima, that the tomb of St. Peter will be moved to Fatima, that the cathedral of the pope (the headquarters of the pope as bishop of Rome) must be destroyed and replaced with one in Fatima. It also holds out the possibility that Rome itself, as a city, will be destroyed, which is something theologians have rejected as impossible for the same reasons, namely that there cannot be a bishop of Rome, i.e. a pope, if the city of Rome does not exist.