The proof of Corollary 3.12 in Mochizuki's IUT papers contains a fundamental gap
3 events · 1 assessment · 1 decision
Structured and assessed
First pass (structure_and_assess), with a Curator suggestion folded in. Structure: created four named arguments. For: "The Scholze–Stix reduction" (new subclaim ebda5df8, the trivial-inequality finding, matcher confirmed novel, seeded 0.75; plus existing Dupuy claim 6311b107 attached contradicts within the line, since it disputes the reduction's target) and "Independent reception" (existing 3f43f48e community non-acceptance and a4ef16cb Joshi's conclusion, both supports). Against: "Mochizuki's rebuttal" (adopted the Curator's suggested contradicts edge to d46c3ddf, completing the two-way structure already present on that claim) and "Formal publication" (new subclaim 72051c4c, PRIMS publication, matcher confirmed novel; scored 0.15/0.1 as settled fact, left a deferred stub). All four arguments given written forms and evaluations. Assessment: SUPPORTED, confidence 0.75, credence 0.85, marginal yield 0.3. Evidence: Scholze–Stix report, Scholze's zbMATH review (post-publication), Joshi's 2024–2025 reports (independent corroboration of incompleteness from a critic of Scholze–Stix's own reasoning), Dupuy–Hilado's explicit non-claim of a proof, broad community non-acceptance, versus Mochizuki's contested rebuttal (credence 0.15 on its page) and the PRIMS publication (weak institutional evidence given venue). Chose supported over contested per §18: no independent expert affirms the proof's completeness after thirteen years, so parity would be false; chose supported over verified because the assessment rests on expert reception and the disputants' reports, not independent verification of the mathematics, and a formal refereeing process concluded otherwise. Recorded three instances read during the pass (zbMATH review affirming, Joshi final report affirming with qualification, Scholze–Stix report affirming the reduction subclaim). Canonical form kept: thirteen words, neutral, recognized by both sides. Importance confirmed at 0.8, contestation 0.8. Escalated to Curator a suggested contradicts edge from the abc-validity claim 188f842c, which currently lacks any link to this, its principal counter-consideration. Marginal yield 0.3: a stronger pass could digest Joshi's November 2025 status report and Dupuy's series in technical detail, but that bears mostly on the trivial-inequality subclaim; the verdict here is unlikely to move until the world does (formalization, independent verification, or independent endorsement of a rebuttal).
Assessed Supported
verdict confidence 0.75 · credence 0.85
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.
Claim entered the graph