Fr. Hunwicke asks...

How to respond to a Pope who is demanding assent to something clearly contrary to the Catholic Faith.

See here for the discussion.

Here is my answer... If I were a convicted Roman Catholic and confronted with a Pope who was, at the least, a material heretic intent on subverting the Deposit of the Faith my answer would be that you resist. You follow the lead of the great French Archbishop Marcel Lefebvre. That is to say you commemorate him at mass, you pray for him ceaselessly, you obey him in all matters not contrary to the faith (also excluding changes to immemorial discipline intended to undermine Catholic Faith), and in all other matters you ignore him until he either repents or God calls him to judgement.That may in some instances require some adjustment in your relationship with your local ordinary. Consider Byzantine Rite parishes or the SSPX. The bottom line is you do what you need to do.

As an Orthodox Christian obviously my answer is in fact different, but that's my advice to Catholics who aren't ready to consider other possibilities. And for the record, I also agree with Fr. H in his rejection of sedevacantism. A Catholic cannot logically subscribe to that without accepting that Rome has fallen. Once you cross that bridge the game is over.