Let me begin by saying most people who I know who "believe the Siri thesis" do not actually claim that they are 100% sure that this is the case, only that it is highly probable that this or some variation of it is correct which Canon Law gives us sufficient room to do.
A few points do come to mind regarding what has been posted so far.
I defer to the theologian Cardinal Billot, the Doctor St. Alphonsus De Liguori, and the great Benedictine Abbot, Prosper Guéranger. They give the following rule: "The peaceful and universal acceptance of a pope by the whole Church is a sign and effect of a valid election."
This was written by veteran Vatican correspondant Gabriella Montemayor.
(1912-2005), whose career spanned 50 years, summarized the rumors that circulated among informed journalists in October 1958:
"Siri was alleged to have been elected at the conclave of 1958, from which, instead, came out Roncalli. The three well-known smoke signals, white, black, and then, finally, white, had aroused not a little perplexity and the same comment throughout the whole of the Italian peninsula: Who had been elected at the first white smoke?"...
"Everyone in Genoa insisted, even from the first day: It most certainly was Siri. Could he have abdicated? Had he been forced out? Was it politics or the Holy Ghost? The mystery remains yet today. "I've talked to several people who remember the white smoke, hek my parents, both of them remember it. One fellow told me his grandfather told him that they hijacked the Church in 58 when the white smoke billowed from the Sistine Chapel and they announced that a new pope had been elected only to declare that it was a mistake an hour later.
Does this sound like "peaceful acceptance of the whole Church" to you? Those in the know knew something was fishy right from the beginning. Even the the articles that appeared in the newspaper alluded to fraud "I guess the stove had a will of it's own" was one of the comments I recall coming form a major Italian newspaper the next day.
Anacletus II was nearly universally accepted by the faithful for several years after his pontificate began, that didn't stop St. Bernard from investigating the situation rallying behind Innocent II once he was convinced that he was the true pope..