What was your prompt? 
1. Can you compare what Fr. Muller says in the PDF to the Monograph we developed?
2. Does Muller interpret Pius IX as saying that the invincibly ignorant have no possibility of salvation or eternal life?
3. Yes, regarding this extraordinary intervention that he speaks of, if the person could be cured of his ignorance through an extraordinary intervention then wouldn't that mean, logically, that his ignorance was VINCIBLE not INVINCIBLE?
4. But Pius IX says nothing about miraculous interventions. Pius IX speaks the requirement that the Invincibly Ignorant observe the natural law in an honest way and by doing that they will attain eternal life by virtue of divine grace and light. If Pius IX simply meant that a miracle must happen, he never implied such.
5. Not only does Pius IX not mention a miraculous intervention, he specifically says that the person will not be held culpable and receive punishment. Well, if there was a miraculous intervention before the person's death, of course the person would not be culpable or receive punishment because he would join the Church. Muller's interpretation makes nonsense of Pius IX's words.