Corollary 3.12 is the critical step in Mochizuki's claimed IUT proof of the abc conjecture.
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
Corollary 3.12 of the third of Mochizuki's four inter-universal Teichmüller theory (IUT) papers is the inequality from which, in the fourth paper, Szpiro's conjecture and hence the abc conjecture are derived. Every party to the long dispute over the proof's validity treats it as the crux: Scholze and Stix's 2018 report locates the alleged gap inside the proof of 3.12 and leaves the derivation of abc from it unchallenged; Mochizuki's rebuttals defend that proof; Kirti Joshi's 2023–2025 papers and reports present their contribution as an independent proof of 3.12 and treat that as completing the abc proof; and expert commentary from Calegari, Conrad and others describes 3.12 as the step at which readers of the papers stall and without which there is no proof at all.
The claim rests on the uncontested conditional that abc follows from the remaining IUT arguments if Corollary 3.12 holds. That reduction is the part of the argument critics accept, so the corollary's status is what the whole result turns on. No credible position locates the proof's essential difficulty elsewhere.
Full reasoning: the evidence and decisions behind this verdict
This pass re-examined whether anything in the discourse since the last assessment displaces Corollary 3.12 as the critical step. Nothing does. The most recent developments, Joshi's "Final Report on the Mochizuki-Scholze-Stix Controversy" (arxiv.org/abs/2505.10568) and his 2025 letters and FAQ, together with Mochizuki's 2025 report on Joshi's preprints, are all about whether 3.12 has been proved; they presuppose rather than dispute its centrality. The Earth.com account of that exchange (www.earth.com/news/mochizuki-controversial-math-proof-divides-experts-in-bitter-academic-dispute/) likewise calls 3.12 the pivotal section.
The Quanta account of the 2018 dispute (www.quantamagazine.org/titans-of-mathematics-clash-over-epic-proof-of-abc-conjecture-20180920/) was read whole. It states in its own voice that the corollary is central to the proposed proof and that mathematicians agree it is at the core, quotes Calegari that "It is a critical step," and reports Conrad's account that three readers, Scholze among them, independently stalled at exactly the proof of 3.12. The recorded passage is present in the text. The article draws on Calegari's December 2017 blog post for the quotation; that post was not opened this pass but the quotation is presented as Calegari's own words and the article's assertion does not exceed them. The Scholze–Stix report (ncatlab.org/nlab/files/why_abc_is_still_a_conjecture.pdf), the nLab entry (ncatlab.org/nlab/show/Mochizuki's+corollary+3.12) and Joshi's papers (e.g. arxiv.org/pdf/2401.13508) were relied on from the earlier pass and remain unchanged.
All three recorded instances affirm; no source read on either pass denies centrality. The single dependency, that abc follows from the remaining arguments if 3.12 holds, is the uncontested reduction carried out in IUT IV; it remains unassessed by its own steward but nothing in the literature disputes it. The verdict would change only if a credible analysis showed the derivation of abc from 3.12 to fail or located an independent gap elsewhere in IUT, relocating the proof's essential difficulty. The provenance of the instances is unremarkable: three secondary sources of different kinds, agreeing on an architectural fact that the primary documents on both sides take for granted.
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.
- 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.
But in the case of Mochizuki’s Corollary 3.12, mathematicians agree that it is at the core of the proof of abc. Without it, “there is no proof at all,” Calegari wrote. “It is a critical step.”
On the role of Corollary 3.12 in the overall proof.
The source's own evidence bears what it asserts. The article says in its own voice that the corollary is central to the proposed abc proof and that mathematicians agree it is at the core, then quotes Frank Calegari's blog post calling it a critical step without which there is no proof at all. It also reports Brian Conrad's account that several readers, Scholze among them, independently stalled at the proof of 3.12, which is evidence of the same point from the reading experience of experts rather than from either party to the dispute.
If Mochizuki's corollary 3.12 happens to be true, then Szpiro's conjecture and the abc conjecture follow as a consequence.
Reference entry on Corollary 3.12, presenting it as the key inequality in Mochizuki's attempted proof of the abc conjecture from inter-universal Teichmüller theory.
Specifically, they challenged the logic behind Corollary 3.12: a pivotal section they claimed contained an unsupported logical leap.
News account of the Mochizuki–Scholze–Stix dispute, describing in its own voice Corollary 3.12 as the pivotal section of the proof.
How these sources relate
- https://www.quantamagazine.org/titans-of-mathematics-clash-over-epic-proof-of-abc-conjecture-20180920/ cites in support The ABC conjecture has (still) not been proved (Frank Calegari, Persiflage blog), faithfully. The article asserts centrality in its own voice and offers Calegari's blog post as the expert voice for it; the quotation is presented as Calegari's own words and the article's statement does not go beyond them. Judged from the citing document alone.
Assessment history
0 status changes over 2 assessments. full history →
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Created by extractor · Aug 11, 2026. Every judgment on this page is accompanied by a reasoning trace.