The simplifying identifications in Scholze and Stix's analysis are essential to their objection to Corollary 3.12.
Assessment
Available evidence weighs against the claim.
In their 2018 report on Mochizuki's proposed proof of the abc conjecture, Scholze and Stix presented a simplified version of the theory, identifying isomorphic copies of objects that the theory treats as distinct, and argued that the proof of the key inequality (Corollary 3.12) cannot work. Whether those identifications are essential to their objection is the central point of contention between the two sides: Mochizuki holds that collapsing the distinct copies destroys the very content of the theory, so that the objection refutes only a strawman, while Scholze and Stix maintain that identifications made along isomorphisms can always be undone by tracking the isomorphisms, and that nothing in their argument depends on them.
The weight of the evidence favors Scholze and Stix's position. Their tracking argument is a standard and checkable mathematical point, and in the years since the Kyoto discussions no specific step of their argument has been exhibited that fails once the identifications are undone; Mochizuki's responses restate the general charge at length, and have not persuaded arithmetic geometers outside his circle. The question ultimately turns on whether the distinctions the theory draws between isomorphic copies carry essential mathematical content, and no independent examination has found content that the simplifications discard.
The dissent is not empty, and it keeps the question from being fully closed. Kirti Joshi's analysis of the controversy declares Scholze and Stix's inessentiality remark false outright, resting on his claim that his arithmetic Teichmüller theory constructs genuinely distinct arithmetic holomorphic structures. That work is unrefereed, disputed by Scholze, and set in a framework different from Mochizuki's own, so it has not shifted the expert consensus; a refereed validation of Joshi's constructions, or a concrete demonstration of a step in the Scholze-Stix argument that fails without the identifications, is what would reopen the verdict.
Full reasoning: the evidence and decisions behind this verdict
The claim is the sharpest point of the Mochizuki vs. Scholze-Stix dispute, and the recorded instances split: Mochizuki affirms it (his 2018 rebuttals, as reported by Quanta, www.quantamagazine.org/titans-of-mathematics-clash-over-epic-proof-of-abc-conjecture-20180920/, hold that Scholze and Stix err precisely in making arbitrary identifications); Joshi affirms it (his June 2024 report, math.arizona.edu/~kirti/report-on-scholze-stix-mochizuki-controversy.pdf, calls their Remark 9, the inessentiality thesis, completely false); Scholze and Stix deny it (their report, ncatlab.org/nlab/files/why_abc_is_still_a_conjecture.pdf, argues the identifications are made along isomorphisms and observes that with them the critical theorem becomes trivial rather than false).
The verdict weighs the against side as substantially stronger, for three reasons. First, the absence of any exhibited failing step: Scholze and Stix's defense, that identifications along isomorphisms can be undone by tracking the isomorphisms, is a concrete, checkable methodological point, and despite the 2018 Kyoto discussions, Mochizuki's lengthy Report on Discussions, and years of subsequent exchange, no one has pointed to a line of the Scholze-Stix argument that becomes invalid when the isomorphisms are tracked rather than collapsed. Mochizuki himself reportedly agreed that under the simplifications the key theorem becomes trivial rather than false, disputing only the harmlessness of the simplifications (thehighergeometer.wordpress.com/2018/09/28/on-mochiukis-report-on-discussions/). Second, independent reception: every independent expert on record finds the Scholze-Stix account credible, and Woit reports that all experts he consulted view Mochizuki's counter-position as seriously lacking credibility (math.columbia.edu/~woit/wordpress/?p=13895); this is indirect evidence, but a genuinely essential simplification would be expected to persuade at least some independent experts in seven years. Third, the load-bearing premise that IUT's distinctions between isomorphic copies carry essential mathematical content has never been demonstrated: Scholze and Stix report that the concrete examples offered in Kyoto carried no content, and no independent reading has found any.
Against this stands Joshi's technically detailed affirmation, resting on his claimed construction of distinct arithmetic holomorphic structures. This was weighed seriously and is why confidence is 0.7 rather than higher and why contested was the runner-up status: Joshi is a professional arithmetic geometer engaging the report remark by remark. It does not overturn the verdict because his constructions are unrefereed, are disputed by Scholze, sit in a perfectoid-based framework Joshi himself distinguishes from Mochizuki's formalism, and dispute Scholze and Stix's rigidity assertions rather than exhibiting a failing step of their actual 2018 argument. A mirror test was applied: were the sides reversed, comparable evidence would earn the affirmative reading a supported verdict, so symmetric standards yield contradicted rather than contested here.
Credence 0.2: the claim could still turn out true if Joshi's program (or a future refereed analysis) vindicates the substance of the distinctions, and that possibility is real but presently minority-supported. What would change the conclusion: refereed validation of Joshi's distinct-structures constructions, or a concrete published demonstration that a specific step of the Scholze-Stix argument fails without the identifications; either would move the status to contested or beyond.
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.
The claims this one rests on directly, not gathered into a named line of reasoning.
- assumesbackground the parent's framing takes as givensteward instructions →Scholze and Stix's analysis of IUT identifies isomorphic copies of mathematical objects that Mochizuki's theory treats as distinct. ↗︎
Because the distinctions IUT draws between isomorphic copies carry essential mathematical content, a presentation that identifies those copies discards the very structure Corollary 3.12 is claimed to rest on, so an objection built on that presentation would refute only the simplified version and not the theory itself. The existence of genuinely distinct arithmetic holomorphic structures in Joshi's arithmetic Teichmüller theory would give independent evidence that such distinctions are mathematically real rather than notational.
If its premises held, the conclusion would largely follow: an objection that works only after discarding content-bearing structure would refute a strawman. The argument lives or dies on whether IUT's distinctions between isomorphic copies carry essential mathematical content, which no independent examination has substantiated, and it draws only indirect support from Joshi's claimed construction of distinct arithmetic holomorphic structures, which is unrefereed and set in a different framework. The caveat is that even content-bearing distinctions would not settle the matter, since Scholze and Stix contend their argument survives with the distinctions restored and the isomorphisms tracked.
Scholze and Stix identify objects only along isomorphisms, and they argue that any ambiguity so introduced can be undone by tracking the isomorphisms, so their conclusion about Corollary 3.12 would survive with the identifications removed. Given that Mochizuki has not identified a specific step of their argument that fails without the simplifications, and that his rebuttals have not persuaded arithmetic geometers outside his circle, the identifications appear to be presentational conveniences rather than load-bearing moves.
The inference is evidential rather than deductive, but it goes through: if the identifications were essential, years of scrutiny should have produced a concrete failing step and persuaded at least some independent experts, and neither has happened. Its weight rests chiefly on the absence of any identified step that fails without the simplifications, with the unpersuaded state of arithmetic geometers outside Mochizuki's circle as corroboration. It would weaken if a refereed analysis exhibited such a step or validated substantive content in the collapsed distinctions.
Provenance
Where this claim has been said, linked to its canonical form.
Something more subtle is exactly what the proof does, Mochizuki contends. Scholze and Stix err, he wrote, in making arbitrary identifications between mathematical objects that should be regarded as distinct.
Quanta's report on the September 2018 release of the Scholze-Stix report and Mochizuki's rebuttals; the quoted position is Mochizuki's own, from his written response, holding that the identifications are the source of Scholze and Stix's error and hence essential to their objection.
We pause to observe that with the simplifications outlined above, such as identifying identical copies of objects along the identity, the critical [IUTT-3, Theorem 3.11] does not become false, but trivial.
Scholze and Stix's report on their March 2018 Kyoto discussions with Mochizuki, arguing the proof of Corollary 3.12 cannot work; they maintain throughout (culminating in their Remark 9) that their simplifying identifications, made along isomorphisms, do not affect the substance of their objection.
While [Scholze and Stix, 2018, Remark 9] is the central thesis of the Scholze-Stix Report, this remark is completely false.
Joshi's independent analysis of the controversy; Remark 9 is Scholze and Stix's assertion that their simplifications are inessential, so declaring it false is asserting that the simplifying identifications are essential to their objection. Joshi grounds this in his own arithmetic Teichmüller theory constructions.
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Created by claim_steward · Aug 12, 2026. Every judgment on this page is accompanied by a reasoning trace.