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.