Scholze and Stix's objection to Mochizuki's IUT proof rests on wrongly identifying objects the theory treats as distinct.
Assessment
Credible evidence or argument exists on multiple sides.
This is Shinichi Mochizuki's central rebuttal to the 2018 report in which Peter Scholze and Jakob Stix argued that the proof of Corollary 3.12, the key inequality in Mochizuki's inter-universal Teichmüller theory (IUT) proof of the abc conjecture, does not work. The factual half of the charge is not in serious dispute: Scholze and Stix's analysis does identify isomorphic copies of objects that Mochizuki's formalism keeps distinct, and they say so themselves, presenting these identifications as harmless simplifications. The dispute is entirely over whether those identifications are essential to the objection. Mochizuki maintains that they are, and that collapsing the distinct copies is precisely what creates the apparent contradiction. Scholze and Stix respond that identifications made along isomorphisms cannot change the content of the argument, that any bookkeeping they discard could be restored without affecting the conclusion, and that the examples Mochizuki offered in their week of discussions in Kyoto carried no substance.
The disagreement has never been resolved by a decisive mathematical demonstration either way, and in that sense it remains a genuine standoff between credible parties. But the two sides have not fared equally in independent scrutiny. Since 2018, essentially no arithmetic geometer outside Mochizuki's circle has endorsed his rebuttal, while Scholze's position that the proof of Corollary 3.12 contains a fundamental gap has become the working view of most of the field; Scholze reaffirmed it in his review of the published IUT papers. Kirti Joshi's more recent work complicates the picture from a third direction, arguing that distinct arithmetic holomorphic structures of the kind Mochizuki invokes do exist in a suitable framework, while agreeing that Mochizuki's own papers do not establish them, so the Scholze–Stix objection was justified against the proof as written.
What would resolve the question is what has been missing since 2018: a precise demonstration, in language both sides accept, either that maintaining all of Mochizuki's distinctions blocks the Scholze–Stix argument or that the argument goes through with the distinctions maintained. Until then the claim stands as a live but lopsided dispute: asserted by Mochizuki and his close colleagues, denied by Scholze and Stix, and found unpersuasive by nearly every independent expert who has engaged with the exchange.
Full reasoning: the evidence and decisions behind this verdict
The claim is Mochizuki's charge, made in his 2018 "Report on Discussions" and summarized in Quanta's reporting (www.quantamagazine.org/titans-of-mathematics-clash-over-epic-proof-of-abc-conjecture-20180920/), that Scholze and Stix err by making arbitrary identifications between mathematical objects that should be regarded as distinct. The claim's two load-bearing premises pull in opposite directions. The first, that Scholze and Stix's analysis identifies isomorphic copies IUT treats as distinct, is effectively conceded: their report explicitly simplifies by identifying objects up to isomorphism. The second, that these identifications are essential to the objection, is the crux, and the direct evidence weighs against it. In their report (www.math.columbia.edu/~woit/szpirostillaconjecture.pdf) Scholze and Stix address the charge head-on: identifications along isomorphisms make no difference "up to equivalence of groupoids", any non-commutativity introduced by identifying fundamental groups via isomorphisms could be tracked and undone, and the concrete examples Mochizuki offered in Kyoto "carried no actual content" in their judgment. Mochizuki has never produced a demonstration, accepted by anyone outside his circle, that the objection fails once the distinctions are maintained.
The instance record shows credible assertions on both sides: Mochizuki affirms (via the Quanta account of his rebuttal), Scholze and Stix deny (in their report). That, plus the absence of any formal adjudication, is what keeps the verdict at contested rather than contradicted; contradicted was the runner-up status, and the low credence reflects how lopsided the informed reception has been. Supporting that reading: Scholze's zbMATH review of the published papers maintains that the argument for Corollary 3.12 "is not a proof" (discussed at math.columbia.edu/~woit/wordpress/?p=12775), and no independent arithmetic geometer has publicly endorsed Mochizuki's rebuttal in the seven years since. Kirti Joshi's arithmetic Teichmüller space program (arxiv.org/pdf/2403.10430 and his 2025 "Final Report", arxiv.org/pdf/2505.10568) gives partial comfort to the distinctness intuition, holding that Scholze and Stix asserted incorrectly that distinct arithmetic holomorphic structures cannot exist, but Joshi equally holds that Mochizuki's papers fail to establish their existence, so his work does not vindicate the claim that the objection rests on a wrongful identification; it relocates the failure to Mochizuki's own lack of adequate language.
If the Corollary 3.12 gap claim is eventually assessed as verified, this claim should move toward contradicted; if someone produces a rigorous argument, accepted beyond RIMS, that maintaining the distinctions blocks the Scholze–Stix reduction, it should move toward supported. The credence of 0.15 reflects that the only parties asserting the claim are its author and close colleagues, that the direct technical response to it has stood unrebutted in independent eyes, and that even the most sympathetic third-party program (Joshi's) declines to endorse it as stated.
Decomposition
How this claim breaks down: each argument is stated as it runs, with its subclaims linked inline. ↗︎ opens a subclaim; the map shows how they fit together.
Because Scholze and Stix's analysis identifies isomorphic copies of objects that Mochizuki's theory treats as distinct, and because those identifications are essential to their objection to Corollary 3.12, the objection is an artifact of collapsing distinctions the theory depends on rather than a flaw in the theory itself.
The inference is sound in structure: if the identifications were made and the objection depends on them, the objection does rest on collapsing distinctions. The first premise, that such identifications were made, is effectively conceded by all parties, so the argument lives or dies entirely on whether the identifications are essential to the objection, which Scholze and Stix deny with a specific counter, that identifications along isomorphisms can be tracked and undone without changing the conclusion. A further caveat: even granting essentiality, the charge that the identifications are wrongful assumes the distinct copies carry mathematical content beyond bookkeeping, which is itself the heart of what is disputed.
If the proof of Corollary 3.12 contains a fundamental gap, then Scholze and Stix's objection is substantively correct and does not rest on any illegitimate identification; and given that Mochizuki's rebuttals have not persuaded arithmetic geometers outside his circle, the charge of wrongful identification has failed to withstand independent scrutiny.
Both strands go through. If the proof of Corollary 3.12 contains a fundamental gap, the objection is substantively right and the charge against it collapses; that premise is itself the master question of the dispute and remains unresolved, so the argument's force is real but not decisive. The reception strand, that Mochizuki's rebuttals have not persuaded independent arithmetic geometers, is well supported and provides indirect but meaningful evidence, since a correct rebuttal circulating for seven years among motivated experts would be expected to find some independent endorsement.
Provenance
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Scholze and Stix err, he wrote, in making arbitrary identifications between mathematical objects that should be regarded as distinct.
Mochizuki's rebuttal to Scholze and Stix's objection.
Mochizuki claims that by replacing X by π1(X), things can happen that cannot otherwise happen. Examples are given concerning the action of π1(X) on certain associated monoids. We discussed this at very great length in Kyoto, but none of these examples carried any actual content.
Scholze and Stix's report on their Kyoto discussions with Mochizuki, responding directly to his charge that their objection depends on illegitimate identifications: they argue identifications made up to isomorphism make no difference to the content of the argument and that Mochizuki's counter-examples carried no content.
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Created by extractor · Aug 11, 2026. Every judgment on this page is accompanied by a reasoning trace.