According to the theory, though, he did accept. Paul Williams (the former FBI consultant) indicated that he accepted and took the name Gregory XVII in the 1958 conclave.
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Without opening this entire discussion again, can I suggest that you
read a little of that book on Archive (available with a free account) and just get a sense of it? It is not even remotely credible. Paul Williams is a hardcore leftist and is trying to smear the Church for not being leftist enough, basically. He also considers the Church a completely secular institution, and one that is completely corrupt. He considers it basically a large criminal gang (!).
As far as Williams' claim to be traditional Catholic on some level (I think he didn't use that term, but he claims to be something along those lines), it is completely false. Just read a few pages out of the book.
I spent a few minutes reading that book in random places. It doesn't take long to get the point of the book. It's just a collection of anti-Catholic urban legends, basically. The only one I can remember now is the story of some evil Benedictine monk, I think, in Serbia during World War 2 who tortured children during the War because somehow it helped the nαzιs. I don't remember much else than that, but that should be enough to make the point.
People say, "But if he hated the Church, why would he repeat the story about Cardinal Siri getting elected?" I think this story does play into his general theme, which is that the Church is just a huge criminal gang, mostly criminal because it is not leftist enough (before Vatican 2, I mean). So to say that a pope was elected but some other criminal element (leftist, in this case), forced him out of power by threats of violence in the conclave fits perfectly into his mindset.
Even if it doesn't fit some general theme, the book is just a collection of ridiculous stories that make the Church look bad.
Really, all you need to do is spend 5-10 minutes reading the book at different random places and you'll figure out what this guy's agenda is pretty fast. It's pretty straightforward.
What is noteworthy, also, is that Williams doesn't add anything to the Siri thesis. All the information he puts in that section of the book is only stuff that was circulating the world for probably decades before the book was published. You'd think that if he had access to secret FBI records, that he would have information that wasn't publicly known about this supposed event, right? Well, obviously he didn't.
Since so much else in the book was a bunch of malicious lies against the Church, I think it's safe to assume his claim to inside information about Cardinal Siri is a lie as well. Personally, my theory is that he didn't think anyone was all that interested in the Siri Thesis, or didn't realize the extent of interest, when he inserted that part into the book. Because of that, he was careless in how he wrote the fake footnotes supporting this claim, so he made the mistake of writing some rather specific details into the fake footnotes that referred to non-existent FBI docuмents. When sedevacantists tried to track down these docuмents, who had an interest in the Siri thesis that he wasn't prepared for, and found out that these docuмents cited in his fake footnotes don't actually exist, he had to do some damage control. He refused to respond to anyone who asks him why they can't look up the supposed docuмents he references on this, or explain why the FBI says they don't exist. He literally hangs up the phone on people who call him to ask about this, and refuses to respond to emails. He refused to respond to Mario Derksen and Silvio Mattachione who both emailed him about it. In the second edition of his book, he removed these footnotes that he got caught making up, and replaced them with a vague reference to "FBI Sources" that couldn't be verified, so he wouldn't have to worry about this anymore.
Please, just read some of this book. It'll really be an eye-opener for you as far as Paul Williams is concerned.