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, the central inequality of the third of Shinichi Mochizuki's inter-universal Teichmüller theory (IUT) papers, is where every party to the dispute over the claimed abc proof locates the essential content of the argument. Mochizuki's fourth paper derives Szpiro-type inequalities, and from them the abc conjecture, from the corollary, so abc follows if Corollary 3.12 holds; conversely, the proof's principal critics, Peter Scholze and Jakob Stix, direct their objection entirely at the proof of 3.12, and Frank Calegari has written that without it there is no proof at all. Sympathetic and independent workers agree: Kirti Joshi's attempted rehabilitation of the proof is organized around establishing 3.12, and reference treatments describe the corollary as the key step. The corollary's centrality is thus common ground between the proof's defenders and its critics; what remains contested is whether its proof is correct, a separate question the graph tracks on its own claims.
Full reasoning: the evidence and decisions behind this verdict
Centrality is asserted, in their own voices, by sources on every side of the validity dispute. The Quanta account of the controversy (www.quantamagazine.org/titans-of-mathematics-clash-over-epic-proof-of-abc-conjecture-20180920/) reports mathematicians' agreement that 3.12 is at the core, quoting Calegari that it is "a critical step," and notes Brian Conrad's observation that 3.12 was where readers of the papers repeatedly stalled. The Scholze–Stix report "Why abc is still a conjecture" (ncatlab.org/nlab/files/why_abc_is_still_a_conjecture.pdf) takes the reduction of abc to 3.12 as given and attacks only the corollary's proof. The nLab entry (ncatlab.org/nlab/show/Mochizuki's+corollary+3.12) states that Szpiro and abc follow if 3.12 is true, and Joshi's papers and reports (e.g. arxiv.org/pdf/2401.13508) treat proving 3.12 as equivalent to rescuing the result. All three recorded instances affirm; no source read for this pass denies centrality or locates the proof's essential difficulty elsewhere.
The one dependency the claim rests on is the conditional that abc follows from the remaining IUT arguments if Corollary 3.12 holds: were that false, 3.12 would be one lemma among many rather than the crux. That reduction, carried out in IUT IV, is the uncontested part of the argument; even the critics accept it. An adversarial check for any credible position that 3.12 is not central (for instance, that the real gap lies in IUT IV or in earlier theory) found none: discussion of effectivity of constants in IUT IV exists but does not displace 3.12 as the critical step. The verdict would change only if a credible analysis showed the derivation of abc from 3.12 to fail, relocating the proof's essential difficulty.
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.
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.
Cite this claim: a formal citation with its evidence attached
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Created by extractor · Aug 11, 2026. Every judgment on this page is accompanied by a reasoning trace.