Christ promised every successor of St. Peter, that which He promised Peter, when He declared: I have prayed for you Simon, that your faith never fail.
But this man from Argentina has failed utterly in the faith.
Therefore, he is not a Succesor of St. Peter, because otherwise Christ would have been either a liar or impotent.
But Christ would never refuse a new man to be Pope, unless the old man still held the office of Pope.
Therefore, Benedict XVI by some canonical error remains Pope.*
Therefore, Bergoglio never was the pope, and all Catholcis must break from communion with him.
* Spoiler alert: Benedict on Feb. 11, 2013 renounced ministry, but Canon 332, section 2 says a Pope must renounce munus.