The proof of Corollary 3.12 in Mochizuki's IUT papers contains a fundamental gap
Assessment
Evidence favors the claim, but the chain is incomplete or the sources are secondary.
Corollary 3.12 of the third IUT paper is the step where Mochizuki's abstract machinery is meant to produce the concrete inequality behind the abc conjecture, and the charge that its proof contains a fundamental gap is the central issue in the long-running dispute over whether that conjecture has been proved. The charge originates with Peter Scholze and Jakob Stix, who studied the papers, spent a week discussing them with Mochizuki in Kyoto in 2018, and reported that under their simplifying identifications the argument yields only a trivial inequality, a problem they judged too severe for small modifications to repair. Scholze maintained the position after the papers' formal publication, writing in his zbMATH review that the argument given for the corollary is not a proof.
The years since have strengthened rather than weakened the charge's standing. The mathematical community broadly does not accept the IUT proof, and no independent expert has verified the proof of Corollary 3.12 in the thirteen years the papers have circulated. Notably, Kirti Joshi, whose own program is sympathetic to Mochizuki's central intuitions and critical of Scholze and Stix's reasoning, likewise concludes that the papers do not establish Corollary 3.12 as written, locating the gap in the papers' failure to prove that the distinct arithmetic structures the argument averages over actually exist.
Two considerations cut the other way. Mochizuki maintains that the objection rests on wrongly identifying objects his theory treats as distinct, a rebuttal that remains contested but has persuaded no independent arithmetic geometer. And the papers were peer-reviewed and published in Publications of RIMS in 2021, though the journal is housed at Mochizuki's own institute and the refereeing has not persuaded outside experts. A narrower dissent comes from Taylor Dupuy, who argues that the simplified statement Scholze and Stix refute is not what Corollary 3.12 actually asserts, though Dupuy's own work treats the corollary as unproven rather than proven.
The evidence therefore favors the claim without settling it beyond dispute: every independent expert who has taken a public position holds that the proof as written is at least incomplete, while the deny side consists of Mochizuki, close colleagues, and an opaque refereeing process. The question would be resolved by an independent verification or formalization of the proof, by broad acceptance of a completion such as Joshi claims, or by a demonstration, accepted outside Kyoto, that maintaining the distinctions Mochizuki insists on blocks the Scholze–Stix reduction.
Full reasoning: the evidence and decisions behind this verdict
Direct evidence for the claim. The Scholze–Stix report "Why abc is still a conjecture" (ncatlab.org/nlab/files/why_abc_is_still_a_conjecture.pdf) presents the technical core: with consistent identifications of copies of real numbers, the key inequality (1.5) must omit the scaling factors and becomes empty. This is recorded as the subclaim that the simplified argument yields only a trivial inequality; what the simplified argument yields is not seriously disputed, only whether the simplifications preserve the original argument's content. Scholze's zbMATH review of the published papers (zbmath.org/pdf/07317908.pdf) states flatly that the argument for Corollary 3.12 is not a proof, and adds that at the critical point the argument is too obfuscated to determine which objects are being compared. Quanta's 2018 account (www.quantamagazine.org/titans-of-mathematics-clash-over-epic-proof-of-abc-conjecture-20180920/) records Stix calling the gap serious and unfixable, and notes Calegari's observation that before the report experts could point to no definitive error, which the report changed.
Independent corroboration. Joshi's "Final Report on the Mochizuki-Scholze-Stix Controversy" (arxiv.org/pdf/2505.10568) concludes that without proving the existence of many distinct arithmetic holomorphic structures there is no way to complete the proof of Theorem 3.11 or Corollary 3.12; this is corroboration from an author who simultaneously argues the Scholze–Stix report itself is invalid (his 2024–2025 status reports, math.arizona.edu/~kirti/report-on-scholze-stix-mochizuki-controversy.pdf), so it is not an echo of the Scholze–Stix position. Dupuy and Hilado's series (arxiv.org/pdf/2004.13228) explicitly does not claim a proof of Corollary 3.12 and treats it as an open statement, even while Dupuy argues the Scholze–Stix simplified argument does not refute it. So the strongest independent critics of Scholze and Stix's reasoning still decline to affirm that the proof is complete, which is the telling pattern: the dissent against the gap charge attacks the route to it, not the conclusion that the papers as written fall short.
Evidence against. Mochizuki's rebuttal that the objection rests on wrongful identifications is assessed contested with low credence; its load-bearing premise, that the identifications are essential to the objection, is assessed contradicted, and Scholze and Stix answered it directly in their report (identifications along isomorphisms can be tracked and undone). Publication in Publications of RIMS (2020 acceptance, 2021 publication) is uncontested as fact but weak as evidence: the journal is edited at Mochizuki's home institute, the refereeing is anonymous and unpublished, and post-publication expert reviews (Scholze's) maintained the objection, an outcome essentially unprecedented for a correct proof of a major conjecture.
Weighing. The instance record on this claim is all affirming (Quanta reporting the Scholze–Stix finding, Scholze's zbMATH review, Joshi's qualified affirmation); the deny side is carried structurally by Mochizuki's rebuttal claim, whose own page records his affirmation of it. The status is supported rather than verified for two reasons: this assessment rests on expert reception and on reading the disputants' reports, not on an independent verification of the mathematics, and a formal refereeing process, however compromised its independence, did conclude the opposite. It is supported rather than contested because thirteen years of scrutiny have produced no independent expert who affirms the proof's completeness, and false parity between one research group plus its home institution and the rest of the field would misstate the state of the argument. One caveat on wording: "fundamental" is read as Scholze and Stix meant it, a gap in the essential mechanism rather than a repairable local slip; Joshi affirms the gap in that sense (the papers lack an essential existence result) while denying that the corollary's statement is beyond rescue in a different framework. The credence of 0.85 rather than higher reflects the unresolved possibility, kept alive by Dupuy's and Joshi's criticisms of the Scholze–Stix reduction, that the specific argument establishing the gap misreads the corollary, even though the incompleteness conclusion is independently reached. The verdict would move toward verified if formalization efforts or a consensus review confirmed the gap, and toward contested or contradicted if an independent verification of the original proof appeared or Mochizuki's rebuttal won independent endorsement.
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.
After a week of face-to-face discussions with Mochizuki in Kyoto in 2018, Scholze and Stix reported that under their simplifying identifications the argument for Corollary 3.12 yields only a trivial inequality, and that because the simplifications do not change the substance of the argument, the proof of the corollary is fundamentally flawed. The force of this line is disputed from within: Taylor Dupuy argues that the simplified statement Scholze and Stix refute is not the statement Corollary 3.12 actually makes, so the reduction would establish the gap only if their reading of the corollary is the right one.
The inference goes through if the reduction is faithful: an argument that yields only a trivial inequality proves nothing, so the finding that the simplified argument yields only a trivial inequality establishes the gap provided the simplifications preserve the original argument's content. That proviso is where the argument is attacked from two directions: Mochizuki holds that the simplifications collapse essential distinctions, and Dupuy argues the statement refuted is not the statement Corollary 3.12 makes. The argument therefore carries strong but not decisive weight; it is the primary basis for the claim, and its caveat is the live remainder of the dispute.
A complete proof of a conjecture of this stature, available since 2012 and scrutinized by motivated experts, would be expected to win independent verification; instead the mathematical community broadly does not accept the IUT proof, and Kirti Joshi, working from an independent program sympathetic to Mochizuki's central intuitions, likewise maintains that the papers do not establish Corollary 3.12 as written. The persistent absence of independent verification, from critics and sympathizers alike, is indirect evidence that the proof as written has a gap.
The inference is inductive but sound: a complete proof of a conjecture of this importance, available for thirteen years to motivated experts, would very likely have found independent verification, so its persistent absence is real evidence of a defect. Both premises stand well: broad community non-acceptance is not seriously disputed, and Joshi's independent conclusion that the papers do not establish the corollary as written is particularly probative because it comes from a critic of Scholze and Stix's own reasoning, not an ally.
Because Scholze and Stix's objection rests on wrongly identifying objects the theory treats as distinct, the apparent gap is an artifact of their simplifications collapsing distinctions the proof depends on, and the original argument for Corollary 3.12 stands as written.
The inference is valid: if the objection genuinely rests on collapsing distinctions the proof depends on, the case for the gap is an artifact and the claim fails. The argument lives or dies on the rebuttal claim itself, which is assessed contested with low credence: its crux, that the identifications are essential to the objection, is currently assessed contradicted, and no independent arithmetic geometer has endorsed the rebuttal in the years since it was made. As it stands the argument identifies the one route by which the claim could collapse, but the premise that route needs is weak.
Because the IUT papers were peer-reviewed and published in Publications of RIMS, after roughly eight years of refereeing and with the editorial board publicly standing behind the result, referees with full access to the argument judged the proof of Corollary 3.12 complete, which weighs against the existence of a fundamental gap.
The premise is settled fact: the papers were peer-reviewed and published in Publications of RIMS. The inference from publication to the absence of a gap is weak in this instance, because the journal is housed at Mochizuki's own institute, the refereeing is anonymous and its substance unpublished, and post-publication expert review maintained the objection rather than withdrawing it. The argument counts, but as institutional rather than mathematical evidence, and it is outweighed by the pattern of independent reception.
Provenance
Where this claim has been said, linked to its canonical form.
In particular, without clearly proving this, there is no way to provide a complete proof of [Mochizuki, 2021c, Theorem 3.11] (let alone [Mochizuki, 2021c, Corollary 3.12]).
Joshi's final report argues that Mochizuki's papers do not prove the existence of the many distinct arithmetic holomorphic structures the proof averages over, and therefore do not contain a complete proof of Corollary 3.12; Joshi simultaneously rejects Scholze and Stix's route to the same conclusion and claims the gap is fillable within his own arithmetic Teichmüller framework.
Scholze and Stix argue that a line of reasoning near the end of the proof of “Corollary 3.12” in Mochizuki’s third of four papers is fundamentally flawed.
Scholze and Stix describe what Stix calls a “serious, unfixable gap.”
Unfortunately, the argument given for Corollary 3.12 is not a proof, and the theory built in these papers is clearly insufficient to prove the ABC conjecture.
Scholze's post-publication review of the IUT papers as they appeared in Publications of RIMS, maintaining after formal publication that the argument for Corollary 3.12 does not constitute a proof.
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Created by extractor · Aug 11, 2026. Every judgment on this page is accompanied by a reasoning trace.