ABC Conjecture

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.

This article records tradition as it has been passed down and reported. Its sources are not yet part of the atlas's verified catalogue.

Every proof has a weakest link, the single step a skeptical reader returns to again and again. In Shinichi Mochizuki's claimed proof of the abc conjecture, that step has a name: Corollary 3.12, buried in the third of his four inter-universal Teichmuller theory papers.

An ordinary corollary follows quickly from a theorem already proved. This one runs nine pages, unusually long for something styled as a quick consequence, and it is the point where Peter Scholze and Jakob Stix, after a week studying the argument directly with Mochizuki in Kyoto in March 2018, concluded the whole edifice comes apart. Their report, Why abc is still a conjecture, does not claim to have found a counterexample to the abc conjecture itself, and does not claim Mochizuki's broader theory is worthless. It claims something narrower and, if correct, just as fatal: that a specific inequality Corollary 3.12 depends on compares two quantities using a scale that has quietly been allowed to shift between one side of the argument and the other, so that what looks like a proof of a strong statement is actually, once the shift is accounted for, a proof of something far weaker and already known.

Stix has explained the objection with an analogy: imagine measuring two lengths with a ruler, only to discover partway through that the ruler itself has been stretched or shrunk by an unknown and uncontrolled amount before the second measurement was taken. The comparison between the two measurements no longer tells you what you think it tells you. Stix reached for a related image of his own, Escher's lithograph of a staircase that climbs endlessly upward and still somehow returns to where it started: each individual step in Mochizuki's construction can look locally sound, the objection goes, while the argument as a whole fails to close.

Mochizuki's own response, posted on his website, is that Scholze and Stix simply have not understood the theory well enough to see why the comparison is valid after all, a position he has held consistently since the report appeared. The Research Institute for Mathematical Sciences at Kyoto University, where Mochizuki works and which publishes the journal that eventually printed his papers, has not treated Corollary 3.12 as a fixed defect requiring a rewritten proof; when the papers were formally accepted in 2020, colleagues there said explicitly that the underlying mathematics had not been changed.

What makes Corollary 3.12 worth naming, rather than leaving the dispute as a vague disagreement between experts, is that it gives the controversy a single, checkable location. Anyone who wants to know whether the abc conjecture has really been proved does not have to evaluate five hundred pages of unfamiliar theory in the abstract. They can go to one corollary, one nine page argument, and ask whether the scale genuinely stays fixed from one side of the inequality to the other. So far, most of the field that has looked has sided with Scholze and Stix: the scale does not stay fixed, and the abc conjecture, published proof or not, remains open.