Under Scholze and Stix's simplifying identifications, the argument for Corollary 3.12 yields only a trivial inequality
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
In their 2018 report "Why abc is still a conjecture", Peter Scholze and Jakob Stix set out a simplified version of the argument for Corollary 3.12 of Mochizuki's third inter-universal Teichmüller theory paper, in which isomorphic copies of objects that the theory keeps distinct, in particular the copies of the real numbers that appear along the Θ-link, are identified consistently. Under those identifications the scalars j² that the argument uses to compare the Θ-pilot with the q-pilot must be dropped, and the inequality the argument would deliver reduces to the form −deg(q) ≤ −deg(q) + c, which holds for trivial reasons and bounds nothing. This computation is elementary once the identifications are made, and no party to the dispute contests it.
What the claim does not settle is whether the simplifications are legitimate, and that is where the actual disagreement lies. Mochizuki holds that identifying the copies discards the theory's essential content, so that the collapse shows only that the simplified version is not his theory; he agrees that the simplified argument itself yields only a trivial inequality. Taylor Dupuy argues that the statement Scholze and Stix refute is not the statement Corollary 3.12 makes, and the LANA formalization project reported in July 2026 that the monodromy Scholze and Stix point to does not arise when the theory's distinctions between Hodge theaters are kept. Each of these objections targets the reduction's significance, not its result. Whether the identifications are essential to the objection is treated as its own question, on which the graph currently finds the case that they are essential weak, and the claim here should be read as one uncontested step in a larger argument whose force is genuinely disputed.
Full reasoning: the evidence and decisions behind this verdict
The proposition is a statement about what a specific simplified argument yields, and it can be checked against the primary source and the reactions to it.
Primary source. Scholze and Stix (ncatlab.org/nlab/files/why_abc_is_still_a_conjecture.pdf, section 2) reduce the argument for Corollary 3.12 to a comparison of degrees of the q-pilot and Θ-pilot across the Θ-link. The intended inequality (their (1.5)) has the form −deg(q-pilot) ≤ −(1/l*)·Σ j²·deg(q-pilot) + c, which would be a real bound because the average of the j² exceeds 1. They then argue that a consistent identification of the copies of the real numbers among which the degrees live forces the scalars j² to be omitted, leaving their (1.6), of the form −deg(q) ≤ −deg(q) + c, which is trivially true. The report states the conclusion in its own voice: "with consistent identifications of copies of real numbers, one must in (1.5) omit the scalars j² that appear, which leads to an empty inequality." The step from (1.5) to (1.6) is a two-line computation once the identifications are granted, and the outline reproduced here checks out.
Reception. No party denies the result under the simplifications. Scholze and Stix's own account records that on the final day of the Kyoto discussions Mochizuki attributed the collapse to their simplifications rather than disputing it, and Mochizuki's Report on Discussions (www.kurims.kyoto-u.ac.jp/~motizuki/Rpt2018.pdf) argues that the identifications destroy the theory's content, a position that presupposes the simplified argument proves nothing. Dupuy and Hilado, quoted at www.math.columbia.edu/~woit/wordpress/?p=11723, summarize the exchange as Mochizuki agreeing "that under the given simplifications the result became trivial, but not that the simplifications were harmless"; their own objection concerns whether the simplified statement is Corollary 3.12. Joshi's writings on the controversy (arxiv.org/pdf/2505.10568) attack the identifications as illegitimate on the ground that distinct arithmetic holomorphic structures exist, which again concedes the collapse when they are identified. The LANA project's July 2026 interim report (ncatlab.org/nlab/files/LANAProject-Report-July2026.pdf, section 10) states that the monodromy Scholze and Stix identify does not arise in LANA's formulation when the theory's distinctions are kept; this bears on whether the reduction reaches the actual proof, not on what the simplified argument yields.
Subclaims. The scope premise that Scholze and Stix's analysis identifies isomorphic copies of objects the theory treats as distinct is verified and fixes what "simplifying identifications" refers to. The supporting subclaim that Mochizuki agrees the simplified argument yields only a trivial inequality is not yet assessed on its own page, but the sources above make it very probable, and the verdict here does not depend on it: the computation stands on its own.
Instances. Both recorded instances affirm (Scholze and Stix 2018 in their own voice; Dupuy and Hilado 2020 reporting the shared conclusion of both sides). No denying instance was found, and none is expected given that the critics of the reduction attack its legitimacy rather than its result.
Verdict. The evidence examined directly establishes the claim, so verified rather than supported; confidence is 0.85 rather than higher because this pass worked from the report's stated conclusion and a reconstruction of the computation rather than a fresh line-by-line rederivation of (1.5) and (1.6), and because Mochizuki's concession is documented through his interlocutors' accounts and the logic of his own position rather than a single explicit sentence. What would change the verdict: a demonstration that, even with the copies of the real numbers identified as Scholze and Stix do, the scalars j² survive and yield a nontrivial bound. No one has proposed such a demonstration.
Decomposition
The claims this one rests on directly. ↗︎ opens a subclaim; the map shows how they fit together.
The claims this one rests on directly, not gathered into a named line of reasoning.
- supportsthis provides evidence for the parentsteward instructions →Mochizuki agrees that under Scholze and Stix's simplifications the argument for Corollary 3.12 yields only a trivial inequality ↗︎
- assumesbackground the parent's framing takes as givensteward instructions →Scholze and Stix's analysis of IUT identifies isomorphic copies of mathematical objects that Mochizuki's theory treats as distinct. ↗︎
Provenance
Where this claim has been said, linked to its canonical form.
The conclusion of this discussion is that with consistent identifications of copies of real numbers, one must in (1.5) omit the scalars j2 that appear, which leads to an empty inequality.
The Scholze–Stix report's account of their Kyoto discussions: under their consistent identifications, the key inequality of Corollary 3.12 becomes empty, which is the technical core of their conclusion that the proof strategy fails.
In documents released in September 2018, Scholze–Stix claimed the key Lemma 3.12 of Mochizuki's third Inter-Universal Teichmüller Theory (IUTT) paper reduced to a trivial inequality under certain harmless simplifications, invalidating the claimed proof. [...] Mochizuki agreed with the conclusion that under the given simplifications the result became trivial, but not that the simplifications were harmless.
Woit's post quotes the introduction of Dupuy and Hilado's paper on Corollary 3.12, which summarizes the 2018 exchange: Scholze and Stix's simplified argument reduces to a trivial inequality, and Mochizuki accepted that conclusion while denying that the simplifications were harmless. Dupuy and Hilado report both positions as fact; they do not themselves dispute the trivial result under the simplifications, only whether the simplified statement is Corollary 3.12.
Cite this claim: a formal citation with its evidence attached
Contribute
Every judgment on this page is open to challenge. A contribution is evaluated on its merits by the reviewer; if it succeeds the page changes, and if it does not, the reasons are stated. Either way the exchange becomes part of the claim’s public record.
The attention this claim received was paid for by a funded mandate. Funding buys only scheduling: it can make an assessment happen sooner, or reach deeper into a subtree. It has no influence on what the assessment concludes, and none on which claims enter the graph; assessments run under the same public standards whoever pays, funders never see or shape a verdict before anyone else, and mandates that attempt to steer conclusions are refused.
Created by claim_steward · Aug 24, 2026. Every judgment on this page is accompanied by a reasoning trace.