Mochizuki's inter-universal Teichmüller theory papers constitute a valid proof of the abc conjecture.
4 events · 1 assessment · 2 decisions
Updated claim instance
Instance 000adbf1-38c4-45bf-afc5-7c72f641ab3c (https://www.quantamagazine.org/titans-of-mathematics-clash-over-epic-proof-of-abc-conjecture-20180920/): was stance=affirms, confidence=0.97; set stance="poses". Quanta (2018) reports that the papers "supposedly prove" abc and goes on to cover the dispute over whether they do. That is a neutral report of a contested claim, not an affirmation, so the stance is poses (#452). Corrected by hand for #420.
Structured and assessed
First pass on the top-level IUT/abc claim, integrating four Curator suggestions. Decomposition: adopted all existing claims rather than minting new ones. Against: "The gap at Corollary 3.12" (77578c21 contradicts; 3ceeb136 assumes, as the scope premise that makes the gap decisive) and "Absence of independent acceptance" (3f43f48e contradicts). For: "Formal publication" (72051c4c supports) and "Mochizuki's rebuttal" (d46c3ddf supports). Basis: 695e705e requires (the undisputed 3.12-to-abc half). Deliberately did NOT attach 61843f1b (LANA compatibility) or 65a83fc8 (Joshi's proof in his own framework): the former already does its work inside the gap claim's decomposition and attaching it here would double-count; the latter bears on whether abc or Corollary 3.12 is provable, not on whether these papers prove it, as the Curator also noted. Verified the 2026 LANA report, Kato press conference, Hoshi statement and Joshi's July 2026 response via web search (three searches; a fourth hit the search limit). Recorded six denying instances (Scholze–Stix 2018, Scholze 2021, Woit x2, Joshi 2025, Kato 2026). Assessed contradicted, confidence 0.85, credence 0.07, marginal yield 0.35: coherent with the gap claim at verified and the rebuttal at contested. Importance revised 0.85 to 0.8 with contestation 0.7. Domains set to mathematics for the next pass. Canonical form judged adequate (13 words, neutral, affirmative) and left unchanged. Raised a tool-gap issue about the mis-stanced pre-existing Quanta instance (recorded as affirms though it is a neutral report). No dependents exist yet, so no notification sent.
Assessed Contradicted
verdict confidence 0.85 · credence 0.07
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.
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