This article records tradition as it has been passed down and reported. Its sources are not yet part of the atlas's verified catalogue.
In August 2012 Shinichi Mochizuki, a mathematician at Kyoto University's Research Institute for Mathematical Sciences, quietly posted four papers to his own website. Together they ran past five hundred pages, building on roughly five hundred more pages of his own earlier foundational work, and they claimed to prove the abc conjecture, a statement about the prime factors of coprime integers that had stood open since Joseph Oesterle and David Masser formulated it in 1985. Mochizuki called the body of work inter-universal Teichmuller theory. Almost no one could read it.
That was not an exaggeration or an insult. Mochizuki's theory introduced its own vocabulary and its own framework, largely disconnected from techniques other number theorists already knew, and after its release the number of mathematicians willing to say they had verified it stayed small, by Ivan Fesenko's own count between 12 and 18, nearly all of them people who had studied directly with Mochizuki or his closest collaborators. For six years the wider field was left in an unusual position: unable to find a specific error, but also unable to follow the proof well enough to be sure there was not one.
That changed, or at least sharpened, in March 2018. Peter Scholze, a Fields Medalist at the University of Bonn, and Jakob Stix, of Goethe University Frankfurt, spent a week in Kyoto going through the argument line by line with Mochizuki and his colleague Yuichi Hoshi. Afterward they wrote up their conclusion in a report bluntly titled Why abc is still a conjecture: a specific step, an unusually long nine page argument for something the papers called Corollary 3.12, contained what they considered a serious and, in their view, unfixable gap. Stix later described the trouble as an inequality where the measuring stick used on each side has quietly been allowed to shrink by an uncontrolled factor, so that the inequality no longer means what it appears to mean; he compared the structure directly to M.C. Escher's drawing of a staircase that looks locally consistent at every step and still fails to close when you follow it all the way around. Scholze put it more simply: past a certain figure in the argument, he said, he was entirely unable to follow the logic, and he still considered the conjecture open.
Mochizuki did not concede the point. On his own website he dismissed the criticism as resting on fundamental misunderstandings of his theory. Two years later, in April 2020, two of his colleagues at the Research Institute for Mathematical Sciences announced at a Kyoto press conference that the papers had been accepted for publication in Publications of the Research Institute for Mathematical Sciences, the institute's own journal, of which Mochizuki himself is chief editor. One of them said plainly that the mathematics itself had not changed in response to the Scholze-Stix report; some discussion of the objection would appear in the published version, but no fundamental alteration had been made. The papers appeared in 2021.
Formal publication did not settle the argument. Kiran Kedlaya of UC San Diego, who had put real effort into checking the work, said publicly that he saw no sign the community's opinion had shifted since 2018. A blunter assessment, from the number theorist Frank Calegari, has circulated widely: the abc conjecture, he wrote, is now a theorem in Kyoto and a conjecture everywhere else.