"mathematical projections" were a series of analytical complex arithmetic computations done by Erasmus which involved looking at the histories of the human and Thinking Machine races, and how these computations pointed to the rise of the Ultimate Kwisatz Haderach.
The projections were based upon the advent of the re-born Thinking Machine Empire; compared with the ferociousness and increased violence of the self-destructiveness of human-kind, and human-kind's use of weapons on other humans. This caused the Machine leaders to believe that a great change or great battle, called by the humans, Kralizec, would soon arise. Erasmus correctly believed that the no-ship, the Ithaca, which had escaped from Chapterhouse with a large crew, was the focal point of his mathematical projections.
Capturing the Ithaca
The two machine leaders believed that the Ultimate Kwisatz Haderach was aboard the Ithaca, and they believed that the super-being could tip the upcoming Kralizec in their favor, if he or she were under Machine control. So Omnius and Erasmus, in their guise of the elderly couple Daniel and Marty, quickly cast out their tachyon net. They failed to capture the ship, as captain Duncan Idaho, and his assitants Sheeana and a ghola of Miles Teg, folded space to another universe. For over two decades the Machine leaders would attempt to capture the no-ship, but were always unsuccessful.
The vessel was suspected by Erasmus because of the fact that the Tleilaxu Master Scytale was aboard the Ithaca, and he had a treasure trove of ghola cells of strong personalities from humanity's past. Over the course of the next 26 years, gholas of such notables as Leto II, Paul Atreides, Jessica Atreides, and others, would be decanted and join the crew.
Ghola of Paul Atreides and Paolo
As a back-up to Erasmus' "mathematical projections", the Machine leaders had a clone of the first known super-being, Paul Atreides, decanted on Bandalong by the Tleilaxu Scribe Uxtal. They named and called this clone, Paolo; and to ensure a cruel personality, had him raised by a ghola of Vladimir Harkonnen, on Caladan. Paolo's training was completed by Erasmus later, when Paolo was brought to Synchrony.
Erasmus' mathematical projections proved to be correct as the no-ship was caught and brought to Synchrony, as well. When the Machine leaders realized that both a ghola of Paul Atreides (from the Ithaca) and Paolo were both alive, Omnius mis-interpreted the "projections", and believed one of them to be the super-being. He had both brought to the capital of Synchrony, the Cathedral, and declared a duel-to-the-death. He believed the stronger would be the Ultimate Super-Being.
As Paolo stabbed Paul, and gave him a near-fatal wound; Paolo won Omnius' duel prize, a large dosage of super-potent melange, called ultraspice. Paolo took a massive dosage of the ultraspice, and soon found himself in an irreversible prescience trance. Paolo ended up starving and thirsting to death.
As Paul came self-healed himself from the brink of death, he revealed that the "projections" were correct, but that none other than Duncan Idaho, was the Ultimate Kwisatz Haderach. The Oracle of Time/Norma Cenva quickly appeared in Cathedral, and confirmed Paul's views, and then banished Omnius to another universe permanently; setting herself up as guardian in that universe.
Erasmus then explained to Idaho and all gathered that indeed, his "mathematical projections" were correct. He also explained how he had intentionally allowed Omnius to interpret the data incorrectly; as Omnius believed that the clone and ghola of Paul Atreides was the key to the ultimate super-being. He declared that Omnius was too staid in his view of humans, as Omnius had still wanted to genocidally kill or dominate mankind.