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Gap in Mochizuki's proof of ABC confirmed by Lean
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A team of mathematicians and computer scientists announced that a gap in Shinichi Mochizuki's proof of the abc conjecture has been confirmed using the Lean theorem prover. At a press conference today, the research group LANA (Lean Analysis of the abc Conjecture) presented their findings after two years of work, concluding that the critical step from Theorem 3.11 to Corollary 3.12 in Mochizuki's Inter-universal Teichmüller (IUT) theory cannot be formally verified as written. This development casts further doubt on the proof, which has been contentious since its release in 2012.
The abc conjecture, a fundamental problem in number theory, concerns the relationship between the sum of two integers and their prime factors. Mochizuki's claimed proof, spanning over 500 pages, has been met with skepticism due to its complexity and lack of acceptance by the broader mathematical community. In 2018, mathematicians Peter Scholze and Jakob Stix raised similar concerns in their report, but Mochizuki has maintained that his arguments are correct. The LANA team's use of Lean, a proof assistant that checks mathematical arguments for logical consistency, provides a new level of scrutiny.
Key facts from the LANA report include the identification of a specific step in the IUT theory that is "unformalizable" in its current formulation. The team emphasized that while Mochizuki's explanation of this point has recently begun to evolve, they reserve final judgment. The report also offers a manageable-length analysis of the critical argument, written for the general mathematical public, making the issue more accessible. The analysis includes a comparison with the 2018 Scholze-Stix report, highlighting similarities in the identified issues.
The implications of this finding are significant for the mathematical community. If the gap cannot be resolved, it would mean that Mochizuki's proof remains incomplete, and the abc conjecture will continue to be an open problem. The LANA team's work demonstrates the power of formal verification tools like Lean in exposing logical flaws in complex proofs. It also underscores the need for transparency and reproducibility in mathematical research.
Reactions from mathematicians have been mixed. Some welcome the formal verification approach, seeing it as a way to resolve long-standing disputes. Others caution that while Lean can identify gaps, it does not necessarily prove that the proof is incorrect—only that it is not fully formalized. Mochizuki himself has not publicly responded to the LANA report, but his supporters argue that the IUT theory is still being refined.
Looking ahead, the LANA team plans to continue their work, potentially updating their analysis as Mochizuki's explanations evolve. The mathematical community will be watching closely to see if a resolution emerges. This development highlights the growing role of automated proof assistants in mathematics, suggesting that future groundbreaking proofs may need to pass formal verification to gain widespread acceptance.
