The proof of Corollary 3.12 in Mochizuki's IUT papers contains a fundamental gap
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
Corollary 3.12 of the third Inter-universal Teichmüller Theory paper is the step at which Mochizuki's argument produces the inequality that implies the abc conjecture, and its proof, the passage from Theorem 3.11 to the corollary, is where every serious examination of the papers has located a missing step. Three independent examinations, by different methods and with different sympathies, have now reached that finding. In 2018 Peter Scholze and Jakob Stix reported, after a week of discussion with Mochizuki in Kyoto, that under their simplifying identifications the argument yields only a trivial inequality; Scholze repeated after formal publication that the argument given is not a proof. Kirti Joshi, a critic of the Scholze–Stix reasoning who has built a rival framework sympathetic to Mochizuki's ideas, maintains that the papers do not establish the corollary as written. In July 2026 the LANA project, a Lean formalization effort led by Fumiharu Kato that worked inside Mochizuki's own framework and put its questions to Mochizuki and his collaborator Yuichiro Hoshi directly over two years, reported that the passage depends on an unproven compatibility between the q-pilot and multiradial constructions and that it had not been able to reconstruct a proof of it; Kato stated that the argument as written is unformalizable. Hoshi had already acknowledged, at the project's March 2026 launch, that many LANA members saw an insurmountable wall at exactly this step and that he had not succeeded in dispelling it. In fourteen years no expert outside Mochizuki's circle has affirmed that the proof is complete, and the mathematical community broadly does not accept it.
The evidence against is of two kinds. The papers were refereed and published in Publications of RIMS in 2021, which is a fact but a weak one here: the journal is edited at Mochizuki's home institute, the refereeing is unpublished, and post-publication expert review maintained the objection. Mochizuki's rebuttal holds that Scholze and Stix reached their conclusion only by identifying objects the theory keeps distinct, and the LANA report gives that rebuttal partial support on one point: it finds that the monodromy inconsistency Scholze and Stix exhibit does not arise in a formulation that keeps the theory's distinctions. But this weakens one route to the gap without closing it, since LANA's own analysis, which preserves those distinctions, still finds the final compatibility unproven. Joshi in turn disputes LANA's diagnosis while reaffirming that the papers do not supply what the proof needs. The pattern across all the dissent is the same: the critics of each account of the gap attack the account, not the conclusion that the published proof falls short.
Two things remain open and should be read alongside the verdict. "Fundamental" is taken here to mean a gap in the essential mechanism, the step on which the entire inequality hinges, rather than a repairable slip elsewhere; that is where all three examinations place it. Whether the gap can eventually be filled is a different question: LANA describes the missing compatibility as not manifestly false and has reserved final judgment because Mochizuki's explanations continue to evolve. A proof of that compatibility, accepted by the formalization team or by independent experts, would show the published proof to have been incomplete rather than unsalvageable, and would call for this claim to be reread in that light; an independent endorsement of Mochizuki's position that no step is missing would reopen it entirely.
Full reasoning: the evidence and decisions behind this verdict
What changed since the previous assessment. The previous verdict was supported, with the stated trigger that it would move toward verified if formalization efforts confirmed the gap. The LANA project's interim report of 16 July 2026 (ncatlab.org/nlab/files/LANAProject-Report-July2026.pdf; also github.com/katobungen/LANA_report_202607) does that in substance. Its abstract states that within a framework reconstructing the passage from Theorem 3.11 to Corollary 3.12 the team isolates a specific compatibility problem at the final stage of the argument, between the construction arising from the q-pilot in its native arithmetic holomorphic structure and the construction obtained through the multiradial procedure, and that the project has not yet reconstructed a proof of the required compatibility. Section 10 compares this with Scholze and Stix and states that the proof of the final numerical inequality hinges on the compatibility labelled (9-1), which is not manifestly false, but of which LANA has no proof. Kato's public summary of the 17 July press conference (x.com/FumiharuKato/status/2078017230537892207) states that the argument as written in the papers is unformalizable, with final judgment reserved because Mochizuki's explanation has recently started evolving. Hoshi's remarks at the project's 31 March 2026 announcement (zen.ac.jp/en/zmc/topics/jwz-o8xr3v6f) acknowledge that many LANA members still feel there is an insurmountable wall in the logic deriving Corollary 3.12 from Theorem 3.11 and that he has not fulfilled his role in conveying it. This is direct testimony from the author's side that, after two years of cooperative questioning, the missing step has not been supplied.
Why this is treated as confirmation rather than as a fourth opinion. The LANA finding is different in kind from the earlier expert reports. It was produced by a team that accepted Mochizuki's framework and its distinctions, that had Mochizuki's and Hoshi's cooperation and documents, and whose stated aim was to verify the proof, not to refute it. Its inability to find the argument for the load-bearing compatibility, and the author's side's inability to supply one on request, is as close to a direct examination of whether the published papers contain the step as is available short of a completed formalization. The three routes to the gap (Scholze–Stix via identification of real-number copies and the trivial inequality; Joshi via the unproven existence of distinct arithmetic holomorphic structures; LANA via the unproven q-pilot/multiradial compatibility) are methodologically independent and all land on the same passage.
How the arguments weigh. The new argument, that the passage depends on an unproven compatibility between the q-pilot and multiradial constructions, now carries the most weight because of its provenance; its subclaim is not yet assessed on its own page, but the report and press conference were read directly for this pass. The Scholze–Stix reduction remains sound in outline but its proviso is now under independent pressure: the LANA report states that the monodromy Scholze and Stix point to does not arise in LANA's formulation and does not obstruct Mochizuki's intended strategy, which is why the monodromy claim has been placed against that argument alongside Dupuy's objection. This lowers the weight of the Scholze–Stix route without lowering the credence in the gap, since the route that replaces it reaches the same conclusion while keeping the distinctions Mochizuki insists on. The independent-reception argument is strengthened: Joshi's 30 July 2026 comments (bpb-us-e2.wpmucdn.com/sites.arizona.edu/dist/4/404/files/2026/07/Comments-on-the-LANA-Project-Report-of-Kato-et-al.pdf) call LANA's conclusion incorrect and side with Mochizuki on why, yet in the same document answer no to whether the papers supply the intrinsic labels the proof requires and identify that as the real point behind Scholze and Stix. Woit's 31 July 2026 commentary (www.math.columbia.edu/~woit/wordpress/?p=15770) concludes that what PRIMS published was a proof with a gap. Mochizuki's rebuttal gains one point from LANA (the monodromy) but loses its general force, because a team that granted its premise still found the step missing. Formal publication is unchanged and unchanged in weight.
Instances. All recorded instances affirm (Quanta 2018 reporting Scholze and Stix; Scholze's zbMATH review; Joshi 2025 and 2026, both qualified; Kato 2026, hedged on finality; Woit 2026). The deny side is carried by Mochizuki's rebuttal claim and its instances; no independent expert has affirmed the proof's completeness in this period, and the LANA report's own positioning is agnostic on the theory's ultimate validity rather than affirming it.
Why verified rather than supported, and the residual. The evidence now includes an examination made from inside the framework, with the author's cooperation, that fails to locate the step, plus a concession from the author's collaborator that the wall has not been dispelled. That is direct evidence that the published proof does not contain the argument, not merely indirect evidence from reception. Confidence in the status is 0.75 rather than higher because LANA explicitly withholds a final verdict, its subclaim has not yet had its own assessment, and this pass read the report's abstract, section 10 excerpts and the surrounding statements rather than the full document; supported remains the defensible alternative reading. The credence of 0.88 blends two readings of "fundamental": under the reading used here (a gap at the essential mechanism), the credence is above 0.9; under the stronger Stix reading (unfixable), it is lower, because LANA describes the compatibility as not manifestly false and Mochizuki's explanation is still evolving. What would change the verdict: a proof of the (9-1) compatibility accepted by LANA or by independent experts would show the gap fillable and require rereading the claim under the weaker sense; a completed formalization failing at that step would move confidence to 0.9 or above; an independent endorsement of Mochizuki's position that no step is missing would move the claim to contested.
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, and the LANA formalization project reports that the monodromy inconsistency Scholze and Stix exhibit arises only when IUT's distinctions between Hodge theaters are collapsed, so the reduction establishes the gap only if their reading of the corollary is the right one and their identifications preserve the original argument's content.
The inference goes through if the reduction is faithful: an argument that yields only a trivial inequality proves nothing, and the finding that the simplified argument yields only a trivial inequality is now established on its own page and conceded by every party, Mochizuki included, so the argument's factual premise is not where any weight is lost. Its whole burden falls on the proviso that the simplifications preserve the original argument's content, which is pressed from three directions: Mochizuki holds that the identifications collapse essential distinctions, Dupuy argues the statement refuted is not the statement Corollary 3.12 makes, and the LANA report finds that the monodromy inconsistency arises only when the theory's distinctions between Hodge theaters are collapsed. The argument therefore remains a credible route to the gap but carries less weight than it once did as the primary basis for the claim; the LANA finding, which keeps the distinctions and still locates a missing step, has taken over the load.
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 fourteen years to motivated experts and now to a cooperative formalization team, 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 was reaffirmed in his July 2026 response to LANA, in which he rejects LANA's diagnosis yet still answers that the papers do not supply what the proof requires. The argument has strengthened, since the dissent against each account of the gap continues to leave the conclusion untouched.
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 Scholze–Stix case for the gap is an artifact. The argument lives or dies on the rebuttal claim itself, which is assessed contested with low credence. The LANA report gives it a partial point, finding that the monodromy Scholze and Stix exhibit does not arise once the distinctions are kept, but the same report, keeping those distinctions, still finds the final compatibility unproven. So even where the rebuttal succeeds against Scholze and Stix's route it no longer rescues the proof, and the argument now weighs against one route to the gap rather than against the gap.
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, post-publication expert review maintained the objection, and a formalization team working with the author's cooperation has since been unable to locate the step the referees must have accepted. The argument counts as institutional rather than mathematical evidence and is outweighed.
The LANA project, a Lean formalization effort led by Fumiharu Kato that worked from within Mochizuki's own framework and put its questions to Mochizuki and Hoshi directly over two years, reported in July 2026 that the passage from Theorem 3.11 to Corollary 3.12 depends on an unproven compatibility between the q-pilot and multiradial constructions, and Kato stated that the argument as written in the papers is unformalizable. Because that compatibility is the step on which the final inequality hinges, and because neither the papers nor two years of direct dialogue with their author supplied a proof of it, the proof of Corollary 3.12 as written has a gap at its core, reached by a route independent of the Scholze–Stix reduction.
The inference goes through: if the step from Theorem 3.11 to Corollary 3.12 rests on a compatibility the papers do not prove, the proof of the corollary as written has a gap at its core. The argument lives on the finding that the passage depends on an unproven compatibility between the q-pilot and multiradial constructions, which is not yet assessed on its own page but rests on the LANA report and press conference read directly for this assessment. Its provenance gives it unusual weight, since the team worked inside Mochizuki's framework with his cooperation and still could not find the argument. The caveat is that LANA calls the compatibility not manifestly false and reserves final judgment, so the argument establishes a missing step in the published proof rather than that the step can never be supplied; Joshi separately disputes LANA's diagnosis while agreeing the papers fall short.
Provenance
Where this claim has been said, linked to its canonical form.
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.”
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.
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.
The way the argument from Theorem 3.11 to Corollary 3.12 is written in the IUT papers is unformalizable. But since Mochizuki's explanation of this point has recently started evolving, we reserve final judgement at this time.
Kato, director of the LANA Lean-formalization project, summarizing the conclusion of the 17 July 2026 press conference on the project's interim report: the argument as written cannot be formalized, with final judgment on the theory withheld while Mochizuki's explanations continue to evolve. Affirms a gap in the proof as written; hedges on whether it is ultimately unfillable.
The LANA people have identified a precise gap in the argument (which for some reason they call a "wall" rather than a "gap"), at the point where Scholze and Stix in 2018 claimed there was a problem. [...] Since no one now has an argument (other than "it's obvious") that fixes the gap, it's clear that what PRIMS published was not a proof, but a proof with a gap, and that the refereeing process failed in this case.
Commentary on the LANA interim report and press conference, concluding in the author's own voice that the published IUT papers contain a proof with a gap at the step Scholze and Stix identified.
This brings us to the important question: does Mochizuki provide such intrinsic labels in (Mochizuki, 2021)? The answer is no. This was the main point on which (Scholze and Stix, 2018) is based
Joshi's response to the LANA interim report. He calls LANA's diagnosis of the gap incorrect and sides with Mochizuki on why LANA went wrong, yet maintains his own position that the published papers do not supply the intrinsic labels (distinct arithmetic holomorphic structures) the proof of Corollary 3.12 requires, so the papers as written do not establish it. A qualified affirmation: agrees there is a gap, disputes both the Scholze-Stix and LANA accounts of it.
Assessment history
1 status change over 2 assessments. full history →
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Created by extractor · Aug 11, 2026. Every judgment on this page is accompanied by a reasoning trace.