This should settle the Sedevacantist question fairly well:
“The inevitable play of human passions, interfering in the election of the Vicar of Christ, may perchance for a while render uncertain the transmission of spiritual power. But when it is proved that the Church, still holding, or once more put in possession of, her liberty, acknowledges in the person of a certain Pope, until then doubtful, the true Sovereign Pontiff, this her very recognition is a proof that, from that moment at least, the occupant of the Apostolic See is as such invested by God himself.” (Abbot Guéranger, O.S.B., The Liturgical Year , Vol XII, pg. 188)
And Cardinal Billot (the great Jesuit theologian of the first half of this century) states:
“Finally, what one may think of the possibility or the impossibility of an heretical pope, there is at least one point absolutely clear which no one can put in doubt, and it is that the acceptance, the adherence, of the Universal Church to a pope will always be, by itself, the infallible sign of the legitimacy of such-and-such a pontiff; and consequently of the existence of all the conditions required for legitimacy.” And this is based on the Church’s attribute of Indefectibility as defined by “the promise of the infallible Providence of Christ [that] ‘the gates of hell shall not prevail against it’ and ‘Behold, I am with you all days even unto the end of the world.’ For the adherence of the Church to a false pontiff would be the same thing as its adherence to a false rule of Faith, since the pope is the living rule of the Faith that the Church has to follow, and that in fact, She always follows.”
He continues:
“God some times can allow that the vacancy of the Apostolic See be for a certain time. He can allow also that a doubt may come concerning the legitimacy of such-and-such an election, but He cannot allow that the whole Church accept as a pontiff one who is not really legitimate. Therefore, from the moment that the pope is accepted by the Church and is united to Her as the head to the body, we can no longer raise the doubt on the possible bias of election or the possible lack of the necessary conditions for legitimacy. Because this adherence of the Church heals in its root all faults committed at the moment of election, and proves infallibly the existence of all the conditions required.”