here’s a good article from Jeff Mirus at CatholicCulture.org: On the lunatic fringe, Francis is not the Pope. Mirus refutes the idea that Benedict is still the Pope, or that Francis could, for any reason, be considered a false pope or antipope. His points:
1. Benedict’s resignation was definitely valid.
2. Francis was duly elected by the Cardinals.
3. “There is absolutely no other way for a pope to lose his office except through death or resignation. He cannot lose it through “heresy”, nor can he lose it because of sin, nor for gross ineptitude.”
4. The Church is indefectible.
You can disagree with Pope Francis. But it is neither faithful nor reasonable to say that Francis is not a valid Pope.
— Ronald L. Conte Jr.