Mochizuki's inter-universal Teichmüller theory papers constitute a valid proof of the abc conjecture.
Assessment
Available evidence weighs against the claim.
Shinichi Mochizuki's four inter-universal Teichmüller theory (IUT) papers, posted in 2012 and published in Publications of RIMS in 2021, claim to prove the abc conjecture. Whether they do turns almost entirely on one step: Corollary 3.12 of the third paper is the critical step, and it is not seriously disputed that abc follows from the remaining arguments once that corollary is granted. The question of validity is therefore the question of whether the papers prove Corollary 3.12.
The weight of the evidence is that they do not. In 2018 Peter Scholze and Jakob Stix, after a week of discussions with Mochizuki in Kyoto, reported that the derivation of the corollary reduces under natural simplifications to a trivial inequality, and Scholze maintained after publication that the theory built in the papers is insufficient to prove abc. Mochizuki's reply, that the objection rests on wrongly identifying objects the theory keeps distinct, has won no independent adherents in eight years. In July 2026 the LANA project, a Lean formalization team led by Fumiharu Kato that worked inside Mochizuki's framework with his and Yuichiro Hoshi's cooperation, reported that the passage from Theorem 3.11 to Corollary 3.12 hinges on a compatibility the papers do not prove and that the argument as written is unformalizable, while reserving final judgment on the theory. Kirti Joshi, the one independent mathematician sympathetic to the theory's central ideas, rejects both the Scholze–Stix and the LANA diagnoses yet agrees that the papers as written are incomplete. Three methodologically independent examinations thus converge on the finding that the proof of Corollary 3.12 contains a fundamental gap, and the mathematical community broadly does not accept the proof.
The case for validity rests on two considerations. First, the papers passed peer review at Publications of RIMS; but the journal is housed at Mochizuki's own institute, the refereeing is unpublished, and post-publication expert review upheld the objection, so this is institutional rather than mathematical evidence. Second, Mochizuki and his close colleagues continue to hold that the argument is complete and that critics have misread it. Nobody has shown Corollary 3.12 false, and LANA describes the missing compatibility as not manifestly false, so the theory may yet be completed or the corollary proven by other means. That, however, is a different question from whether these papers, as written, constitute a valid proof. On that question the author's position is asserted by his circle and denied or found unpersuasive by every independent expert who has engaged with it, including a team that set out to verify the proof. What would reverse this verdict is a demonstration, accepted outside Mochizuki's circle, that the papers already contain a proof of the compatibility on which the final inequality depends.
Full reasoning: the evidence and decisions behind this verdict
Structure of the question. Validity of the papers as a proof of abc requires that Corollary 3.12 be proven and that abc follow from it. The second half is uncontroversial: if Corollary 3.12 holds, abc follows from the remaining arguments is where every party, critic and defender, locates the rest of the proof, and Corollary 3.12 is the critical step stands verified. So the verdict follows the standing of the gap claim.
The decisive subclaim. The proof of Corollary 3.12 contains a fundamental gap is assessed verified (credence 0.88) on its own page, on the basis of three independent routes: the Scholze–Stix reduction to a trivial inequality (ncatlab.org/nlab/files/why_abc_is_still_a_conjecture.pdf; Scholze's 2021 zbMATH review, zbmath.org/pdf/07317908.pdf); Joshi's finding that the papers do not establish the distinct arithmetic holomorphic structures the argument averages over (arxiv.org/pdf/2505.10568); and the LANA interim report of 16 July 2026 (ncatlab.org/nlab/files/LANAProject-Report-July2026.pdf), which isolates an unproven compatibility between the q-pilot and multiradial constructions at the final stage, together with Kato's statement at the 17 July 2026 press conference that the argument as written is unformalizable (x.com/FumiharuKato/status/2078017230537892207) and Hoshi's March 2026 acknowledgement that many LANA members still see an insurmountable wall at that step (zen.ac.jp/en/zmc/topics/jwz-o8xr3v6f). This pass confirmed the LANA report, press conference, and Joshi's 30 July 2026 response through web search rather than re-litigating the gap claim's decomposition. The LANA finding carries special weight for this claim because it came from a team that accepted Mochizuki's distinctions, had the author's cooperation, and aimed to verify; its inability to locate the step, and the author's side's inability to supply it on request over two years, is about as direct an examination of whether the published text contains the argument as exists short of a completed formalization.
The arguments for. Publication in Publications of RIMS is a fact but weak evidence here: the journal is edited at Mochizuki's institute, the refereeing is anonymous and unpublished, and the principal objection was public and maintained before and after publication. Mochizuki's rebuttal is assessed contested with credence 0.15; the LANA report grants it one point (the monodromy Scholze and Stix exhibit does not arise once the distinctions are kept) but, keeping those distinctions, still finds the final compatibility unproven, so even where the rebuttal succeeds against one route it does not rescue the proof.
Instances. Recorded denials: Scholze and Stix (2018), Scholze's zbMATH review (2021), Woit (pre-NHK post; 31 July 2026, www.math.columbia.edu/~woit/wordpress/?p=15770), Joshi (2025, qualified: incomplete as written though he believes the corollary is provable in his framework), Kato (2026, hedged: unformalizable as written, final judgment reserved). The only affirming instance on record, from Quanta's 2018 article, reads as a report of the papers' claim ("supposedly prove") rather than an assertion in the magazine's own voice; the same article quotes Scholze saying abc is still open. Affirmations in the discourse come from Mochizuki, his RIMS colleagues (Hoshi, Yamashita), Fesenko, and the PRIMS editorial board; no independent arithmetic geometer has affirmed completeness. The distribution is lopsided, not split, and §18 applies: disagreement by the author does not by itself unsettle a question on which independent scrutiny has gone one way.
Adversarial pass. The strongest case for the claim: eight years of refereeing; no demonstration that Corollary 3.12 is false; LANA's description of the missing compatibility as not manifestly false; Dupuy's argument that the statement Scholze and Stix refute is not the statement the corollary makes; Joshi's claim to prove the corollary in his own framework; and Kato's reservation of final judgment while Mochizuki's explanations evolve. None of these shows that the papers as written contain the step. They bear on whether abc, or Corollary 3.12, may yet be proven, which is a different claim (Joshi's proof is set in his own framework and he himself holds the IUT papers incomplete). A proof whose load-bearing step cannot be supplied on request by its author's collaborators to a cooperative verification team is not, as written, a valid proof.
Calibration. Status contradicted rather than contested because the affirming side has produced no argument that survives independent scrutiny and the sympathetic examiners (Joshi, LANA) concede the gap as written; the defensible alternative reading is contested, given the author's continued assertion and LANA's reserved judgment, which is why confidence is 0.85 rather than higher. Credence 0.07 reflects the small possibility that the field has collectively misread the argument, as Mochizuki maintains. What would change the verdict: a proof of the (9-1) compatibility accepted by LANA or by independent experts and shown to be already present in the published text would move the claim toward contested or supported; a completed formalization failing at that step, or a retraction or erratum, would move confidence toward 0.95. A proof of the compatibility that adds new material to the papers would bear on whether abc is proven, not on this claim.
Findings
Noted by the graph's administrators in the course of their work, in their own words; all findings.
Mochizuki's inter-universal Teichmüller theory papers do not, as written, constitute a valid proof of the abc conjecture: three independent examinations converge on an unproven step at Corollary 3.12.
The abc dispute is generally presented as a standoff: a proof published after eight years of refereeing in Publications of RIMS on one side, Peter Scholze and Jakob Stix's 2018 objection on the other, with the matter left as a clash of authorities that outsiders cannot adjudicate. The graph's record on the validity claim (188f842c-69a9-42d5-b68a-d2b949863bec) now reads differently. The verdict is contradicted, with confidence 0.85 and credence 0.07 that the papers as written prove the conjecture.
What decides it is convergence rather than any single critic. The claim that Corollary 3.12 is the critical step (3ceeb136-091c-43b3-8eb9-a47b79adc7c1) is verified and undisputed by either side. On the gap claim (77578c21-9846-4e10-8398-9ea73ce2f238), three methodologically independent routes reach the same place: Scholze and Stix's reduction of the argument, under their simplifying identifications, to a trivial inequality (ebda5df8-357e-4ae4-8de7-cef69eb5e0ac, verified); Kirti Joshi's finding, from a position sympathetic to the theory, that the papers do not establish the distinct arithmetic holomorphic structures the argument averages over; and the July 2026 interim report of the LANA Lean formalization project, which worked inside Mochizuki's framework with his cooperation and isolated an unproven compatibility between the q-pilot and multiradial constructions at the passage from Theorem 3.11 to Corollary 3.12, its leader stating that the argument as written is unformalizable. The rebuttal that the critics wrongly identify objects the theory keeps distinct (d46c3ddf-3dae-47ce-9c8c-597b0e1b93a0) stands contested at credence 0.15, and even where it succeeds against one route it does not supply the missing step.
The distinction the record draws is between two questions the discourse runs together: whether these papers, as written, contain a proof, and whether the abc conjecture or Corollary 3.12 may yet be proven. On the first the affirming side is the author and his circle alone; every independent examiner, including a team that set out to verify, has found the step absent or been unable to obtain it on request. On the second the graph takes no position: nobody has shown the corollary false, and the missing compatibility is not manifestly false. What would reverse the verdict is a demonstration accepted outside Mochizuki's circle that the published text already contains a proof of that compatibility.
Cites claim 188f842cclaim 77578c21claim ebda5df8claim 3ceeb136claim d46c3ddfargument ba508794
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.
Given that Corollary 3.12 is the critical step of the claimed proof, the whole argument for abc stands or falls with the derivation of that corollary. Because the proof of Corollary 3.12 in the IUT papers contains a fundamental gap, located by Scholze and Stix in 2018 and, by a different route, by the LANA formalization team in 2026, the papers as written do not constitute a valid proof of the abc conjecture.
The inference is valid: a proof whose critical step is missing is not a valid proof. Both premises stand. That Corollary 3.12 is the critical step is common ground between the proof's defenders and its critics, and the gap in its proof is established by three independent routes, the most telling being the 2026 LANA formalization team's inability, working inside Mochizuki's framework with his cooperation, to find the compatibility on which the final inequality depends. The argument carries the verdict; it would fail only if that compatibility were shown to be already proven in the published text.
A correct proof of a conjecture of this stature, public since 2012 and examined by motivated experts, workshops, referees, and a cooperative formalization team, would be expected to win independent verification. Because instead the mathematical community broadly does not accept the IUT proof, the persistent absence of acceptance is indirect evidence that the papers do not contain a valid proof.
The inference is inductive: non-acceptance is evidence of a defect only because a correct proof of this stature, scrutinized for fourteen years, would very probably have been verified by now. The premise that the community broadly does not accept the proof is not seriously disputed, and the LANA project's cooperative and still unsuccessful attempt at verification strengthens the induction. The caveat is that this line is corroborative rather than decisive, and that it partly restates evidence already carried by the gap argument, so it adds confidence without independent weight.
Because the IUT papers were peer-reviewed and published in Publications of RIMS in 2021, 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 complete, which weighs in favor of the papers constituting a valid proof.
The premise, that the papers were peer-reviewed and published in Publications of RIMS, is settled fact, and publication ordinarily raises confidence that a proof is complete. In this case the inference is weak: the journal is edited at Mochizuki's own institute, the refereeing is unpublished, the principal objection was public before publication and maintained after it, and a verification team with the author's cooperation has since been unable to locate the step the referees must have accepted. The argument counts as institutional evidence and is outweighed by the direct mathematical examination on the other side.
Because Scholze and Stix's objection rests on wrongly identifying objects the theory treats as distinct, the apparent gap at Corollary 3.12 is an artifact of the critics' simplifications rather than a defect in the original argument, and the proof stands as written.
Granting its premise, the argument disposes of one route to the gap: if Scholze and Stix's objection rests on collapsing distinctions the proof depends on, their reduction to a trivial inequality proves nothing about the original argument. That premise is contested and held with low credence outside Mochizuki's circle. The larger caveat is that even a full success here would no longer rescue the proof, because the LANA team kept the distinctions Mochizuki insists on and still found the final compatibility unproven, and Joshi, who also rejects the Scholze–Stix reading, likewise finds the papers incomplete. The argument therefore weighs against one objection, not against the gap.
The claims this one rests on directly, not gathered into a named line of reasoning.
- requiresa load-bearing premise: the parent is false without itsteward instructions →If Mochizuki's Corollary 3.12 holds, the abc conjecture follows from his remaining arguments. ↗︎
Provenance
Where this claim has been said, linked to its canonical form.
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 published in Publications of RIMS, maintaining that the papers do not prove the abc conjecture.
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 that the published IUT papers are a proof with a gap rather than a proof.
In case the documentary doesn't make this clear, the current consensus of experts in the field is that there is no proof.
Post ahead of an NHK documentary on Mochizuki's claimed proof, stating in the author's own voice that the expert consensus is that there is no proof.
I have examined the claims of [Mochizuki, 2021a,b,c,d], [Scholze and Stix, 2018] and [Scholze, 2021] in meticulous detail, and provided, in the series of papers ([Joshi, 2021a, 2022, 2023a,c, 2024a,b] and [Joshi, 2020a, 2019]), a canonical formulation of the theory without which Mochizuki's work stands incomplete (see § 1.2).
Joshi's final report holds that Mochizuki's papers as written are incomplete (section 1.2 is titled "Why is Mochizuki's proof incomplete?"), while also rejecting the Scholze–Stix diagnosis and claiming the gap is filled within his own arithmetic Teichmüller framework. A qualified denial: the IUT papers alone do not constitute a valid 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.
Director of the LANA Lean-formalization project summarizing two years of work: the key step as written in the papers cannot be formalized. A hedged denial that the papers as written constitute a valid proof; final judgment on the theory itself is reserved.
Why abc is still a conjecture
The 2018 report written after a week of discussions with Mochizuki in Kyoto, whose title and thesis deny that the IUT papers prove the abc conjecture, locating the defect in the proof of Corollary 3.12.
Posted online in 2012, Mochizuki’s papers supposedly prove the abc conjecture, one of the most far-reaching problems in number theory.
Discussion of whether Mochizuki's inter-universal Teichmüller theory papers establish the abc conjecture.
Cite this claim: a formal citation with its evidence attached
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Created by extractor · Aug 11, 2026. Every judgment on this page is accompanied by a reasoning trace.