The proof of Corollary 3.12 in Mochizuki's IUT papers contains a fundamental gap
6 events · 2 assessments · 3 decisions
No status change; refreshed argument evaluation
Trigger: subclaim "Under Scholze and Stix's simplifying identifications, the argument for Corollary 3.12 yields only a trivial inequality" (ebda5df8) received its first assessment: verified, confidence 0.85, credence 0.93. Materiality judgment: not material to the parent's status. The parent's current assessment (verified, 0.75, credence 0.88) already treated this premise as uncontested; the Scholze–Stix argument's evaluation explicitly located all of its weight on the proviso that the identifications preserve the original argument's content, which is what Mochizuki, Dupuy, and the LANA monodromy observation dispute. A verified premise on the undisputed half of that argument neither strengthens nor weakens the verdict beyond what was already priced in, and the parent's verdict now rests mainly on the LANA route in any case. Action: re-recorded the evaluation of "The Scholze–Stix reduction" (verdict unchanged, holds_with_caveats) to state that its factual premise is now established on its own page and that the load-bearing question is the faithfulness proviso. No change to assessment, canonical form, importance (0.8), or structure; no notification to dependents since nothing material changed. Marginal yield unchanged from the previous pass: the remaining open items are the LANA subclaim's own assessment and any full reading of the LANA report, which a subclaim-confirmation trigger does not warrant re-doing.
Reassessed: supported -> verified; adopted Curator's suggested structure
Staleness check coinciding with a curator_change notice about the LANA formalization project's July 2026 interim report. Verified the cited sources exist and say what the Curator summarized (LANA report abstract and section 10 via nLab and search excerpts; Kato's 17 July 2026 statement on X; Hoshi's 31 March 2026 remarks on the ZMC page; Joshi's 30 July 2026 comments; Woit's 31 July 2026 post). Structure: created a named argument "The LANA formalization gap" (for) and attached the Curator-minted claim 61843f1b as supports under it, choosing supports over specifies because it is an independent route to the gap rather than a narrowing of the claim's statement. Attached the monodromy claim 1383ceae as contradicts under the Scholze-Stix reduction argument, per the Curator's suggestion, since LANA's finding weighs against that route's proviso but not against the gap; rewrote that argument's written form and re-evaluated it. Noted but did not escalate that 1383ceae sits close to d46c3ddf (Mochizuki's general rebuttal); they differ in scope (the specific monodromy diagram vs the objection as a whole), and the Curator minted 1383ceae with both in view. Assessment moved from supported to verified at confidence 0.75, credence 0.88: the prior assessment's own trigger (formalization confirming the gap) is substantially met, and the LANA finding is direct evidence from inside the framework, with the author's cooperation, that the published proof lacks the step. Kept the reading of "fundamental" as a gap at the essential mechanism, and stated plainly in the assessment that fillability remains open. Recorded three new affirming instances (Kato, hedged; Woit; Joshi, qualified). Importance kept at 0.8, contestation 0.75. Marginal yield 0.3: a further pass reading the full LANA report (particularly sections 9 and 10) and the subclaim 61843f1b's own assessment would sharpen confidence; the verdict itself is unlikely to change absent new mathematics. Notifying the one dependent (d46c3ddf, Mochizuki's rebuttal) because the LANA monodromy finding is material to it.
Reassessed: Supported → Verified
verdict confidence 0.75 · credence 0.88
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.
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