Anyone accepted by the body of Bishops as Roman Pontiff is a valid Roman Pontiff, and no valid Pope can ever lose his validity (but he may validly resign).
St. Robert Bellarmine held that if a man was invalidly elected, but subsequently was accepted by the body of Bishops as Roman Pontiff, he then becomes validly elected and the valid Roman Pontiff. For though the Cardinals have long elected the Roman Pontiff, they are doing so solely as the representatives of the body of Bishops, in whom resides the gift from Christ to choose the successor of Peter. Thus, any Pope accepted by the body of Bishops — regardless of the circumstances of his election, no matter how corrupt it may have been — is the validly elected Roman Pontiff.
Pope Francis has been accepted by the body of Bishops as the Roman Pontiff, therefore he is the valid Pope. And all fables and lies about his election, tending to discredit him or his election, are therefore false accusations against the Vicar of Christ.
This is an important truth to understand. The election of each Roman Pontiff after Peter is entrusted by Christ to the body of Bishops, and does not reside in the Cardinals. The Cardinal electors act solely as the representatives of the body of Bishops. At some point in the future, a Pope or Council or the Cardinals after the passing of the Pope may decide to permit some Bishops to participate in the election of the Roman Pontiff. For the authority exercised by the Cardinals to choose the Roman Pontiff has always been entrusted to them, by Christ certainly, but through the body of Bishops.