ABC Conjecture Update: The LANA Project

[00:00:00]Let me give you a new update on the unfortunate situation that surrounds the ABC conjecture. The result that is a theorem in some parts of Japan and it remains a conjecture elsewhere in the world. And the news today is the Lean A project, an international effort to try to use Lean, a programming language, to formally verify mathematics, to verify Mochizuki's proof of the ABC conjecture.

[00:00:29]The Lean A project got started in 2023 in full swing by 2024 and then several working groups at different universities around the world in Japan, but also in Canada, in the US, and in Europe have been working on this formalization effort of anabelian geometry, which that has been quite successful, and then trying to verify Mochizuki's work, and that has run into a problem. And the problem is a very familiar problem because the formalization effort is stuck exactly at trying to verify, formally verify, the implication that in Mochizuki's work, theorem 3.11, which essentially summarizes IUT, implies corollary 3.12, some very uh delicate inequality of real numbers. And I say this is a familiar roadblock because it's exactly the same spot that is Scholze and Stix back in 2018 identified as a significant problem that was not justified in Mochizuki's work.

[00:01:43]And the Lean A project, in their report just published in July, are being very careful to say that they just don't know whether the implication of 3.11 to 3.12 is false, or it is true, but they don't know enough about the theory to know how one implies the other. And here, it's again a very familiar problem in that from the beginning, Mochizuki has not provided enough details on his own work and has not been able to explain to others why that implication, why much of the work follows, but in particular that implication. And when Scholzen Stix brought this up, Mochizuki replied in an extremely aggressive, offensive, and unprofessional manner to that question about where it how does this implication follow? Now, let me add one more wrinkle to this story because back in 2021, Kirti Joshi from the University of Arizona started publishing some preprints where he claimed he was adding that detail on those missing details in uh Mochizuki that were not present in IUT. And in fact, he claimed he was reworking this uh Teichmüller theory almost from the ground up to be able to conclude the proof of ABC. And in fact, by 2024, Joshi did publish a preprint where he claimed he had completed this construction and the proof of the existence of this arithmetic Teichmüller theory that he says this like the proof of the existence is missing from uh Mochizuki's work, and he claims that in his work, the proof of ABC is actually complete. And very recently, Joshi has published a scathing letter to Katuuti, the leader of the Lana group, because they seem to be completely ignoring Joshi's work, just like Mochizuki, completely ignored and ridiculed, again very unprofessionally, Joshi's work. I should mention that since I posted my previous update back in 2025 on the ABC Conjecture, I had a long video call with Joshi and I should say that Joshi is, first of all, a very respectful, polite, and very capable mathematician whose work has been largely ignored by the community and I think it really deserves more attention than it has received. And I do think the Lean up project owes some attention to Joshi's work and they should either try to verify that Joshi is right or prove that there are gaps in Joshi's work or try to reconcile Mochizuki's work with Joshi's work. Now, I should also mention that I have also talked to other experts that have looked into Joshi's work and they also find it hard to parse. But, however, unlike Mochizuki, Joshi has always been open to talk to anyone who is paying attention to his work to explain and to interact mathematically with other experts. So, I really hope the Lean up project tries to pay some attention to Joshi's work because it may be the key to resolve the problems with Corollary 3.12. And in the letter from Joshi, he explains why there are gaps in Mochizuki and why his work does solve and resolve those issues.