As to your previous statements - I complete disagree.
Reference this article: https://nonvenipacem.com/2017/07/03/count-me-in-moral-certitude-and-the-invalid-abdication-of-pope-benedict-xvi-still-reigning/
In his final general audience, 27 Feb 2013, he said this: (emphasis mine)
Here, allow me to go back once again to 19 April 2005. The real gravity of the decision was also due to the fact that from that moment on I was engaged always and forever by the Lord. Always – anyone who accepts the Petrine ministry no longer has any privacy. He belongs always and completely to everyone, to the whole Church. In a manner of speaking, the private dimension of his life is completely eliminated. I was able to experience, and I experience it even now, that one receives one’s life precisely when one gives it away. Earlier I said that many people who love the Lord also love the Successor of Saint Peter and feel great affection for him; that the Pope truly has brothers and sisters, sons and daughters, throughout the world, and that he feels secure in the embrace of your communion; because he no longer belongs to himself, he belongs to all and all belong to him.
The “always” is also a “for ever” – there can no longer be a return to the private sphere. My decision to resign the active exercise of the ministry does not revoke this. I do not return to private life, to a life of travel, meetings, receptions, conferences, and so on. I am not abandoning the cross, but remaining in a new way at the side of the crucified Lord. I no longer bear the power of office for the governance of the Church, but in the service of prayer I remain, so to speak, in the enclosure of Saint Peter. Saint Benedict, whose name I bear as Pope, will be a great example for me in this. He showed us the way for a life which, whether active or passive, is completely given over to the work of God.
https://w2.vatican.va/content/benedict-xvi/en/audiences/2013/documents/hf_ben-xvi_aud_20130227.html
"Now, combine those words with his decision to retain the papal title as an emeritus, to retain the vesture, to physically remain at the Vatican, etc etc. This is the evidence that is contemporary with the (supposed) abdication. It doesn’t invalidate the other evidence, where he says he is renouncing, but it sure is a serious counterweight to it."
The “Petrine ministry” is not “the papacy.” Ratzinger uses the term “Petrine ministry” precisely because it’s NOT “the papacy.”
If the See is vacant, there is no Pope, because the Pope is the OCCUPANT of the See of Rome. I.e., the Pope is the bishop of Rome, and the bishop of Rome is the Pope.
NOTHING that Ratzinger may think or say about what he is doing NOW (praying, wearing white, etc.) has any bearing whatsoever on the act of resignation. He specified the NATURE of his act of resignation by stating explicitly that its OBJECT would be that “the See will be vacant.”
Ratzinger never said, “My successor will not pray, because I will be praying,” etc. I.e., Ratzinger never said that his successor would NOT exercise the full power and authority and function of Pope, because Ratzinger was going to retain some of it. If Ratzinger had said that, that might call into question his resignation.
Again: Ratzinger NEVER said that his successor would be DEPRIVED of some aspect or exercise of the papacy, on the grounds that Ratzinger was KEEPING IT.
As to whether Bergoglio is Pope at this moment: Only God knows. You (and I) simply cannot make that judgment. As gut-wrenching as the thought may be: God has not entrusted you with a sliver of authority to make judgments of that nature.