The passage from Theorem 3.11 to Corollary 3.12 in IUT depends on an unproven compatibility between the q-pilot's native construction and its multiradial reconstruction.
3 events · 1 assessment · 1 decision
Structured and assessed on curator creation
First pass on a Curator-created claim. (1) Canonical wording checked against the LANA report's abstract (obtained via search; the PDF reader fails on all PDFs, known issue 4fb8e166 joined as a sighting): the Curator's rendering was accurate; tightened to "the q-pilot's native construction and its multiradial reconstruction" to make clear that two constructions of one object are compared. Direction unchanged. (2) Decomposition: linked existing 39b56a7a (LANA's reading of the intended derivation as a within-structure degree comparison) as requires, since the compatibility (9-1) arises from that reconstruction; linked existing 65a83fc8 (Joshi's ATS proof of Corollary 3.12) as contradicts; minted ea1d7030 (argument as written cannot be formalized), seeded 0.7, importance 0.4, as the node hosting the Kato/Mochizuki dispute on formalizability. Did not mint a separate Mochizuki-position node: his denial belongs as an instance on ea1d7030 and on the parent 77578c21. Did not attach 6bb6f98f (same obstruction as Scholze-Stix) as an edge because it is lateral rather than load-bearing; linked in prose. (3) Instances: LANA report (affirms), Kato X statement (affirms, 0.7), Joshi comments (denies, 0.6). Hoshi's remark judged an acknowledgment, not an assertion; nLab and BigGo judged neutral reports. No Mochizuki instance: no verbatim passage denying this claim was found; his April 2026 slides imply formalizability but do not address the July report. (4) Provenance map written and marked material: all affirmations trace to one document. (5) Assessed supported, confidence 0.7, credence 0.8, marginal yield 0.5 (report and Joshi comments unread in full; Mochizuki reply outstanding). Importance 0.5, contestation 0.8, matching the Curator's estimate. Five web searches used.
Assessed Supported
verdict confidence 0.70 · credence 0.80
Corollary 3.12 of the third inter-universal Teichmüller theory paper is the inequality from which Mochizuki derives the abc conjecture, and its proof from Theorem 3.11 is the step every serious examination of the papers has found wanting. This claim locates the difficulty precisely, as the LANA project did in its July 2026 interim report: the argument computes the q-pilot twice, once directly in its native arithmetic holomorphic structure and once through the anabelian and Kummer-theoretic multiradial procedure, and the final inequality goes through only if the two constructions are compatible. The report states that the project, after two years of reconstruction carried out in direct dialogue with Mochizuki and his collaborator Yuichiro Hoshi, had not been able to prove that compatibility, and its director Fumiharu Kato said at the accompanying press conference that the argument as written is unformalizable, while reserving final judgment because Mochizuki's explanations had begun to evolve. The evidence favors the claim. The location of the dependency follows from LANA's reading of the intended derivation as a comparison of degrees within one arithmetic holomorphic structure, a reading developed with the theory's authors rather than imposed from outside, and Hoshi had already acknowledged in March 2026 that many project members saw an insurmountable wall at exactly this logical step and that he had not dispelled it. That the written argument resists formalization at this point is the strongest independent sign that the compatibility is asserted rather than proved in the text. The project is careful to say the compatibility is not manifestly false; the claim is that it is unproven, not that it fails. Two dissents should be read alongside this. Mochizuki has maintained since 2018 that the published proof is complete, embraced Lean formalization in 2025, and presented skeletal code for the step in April 2026; his considered reply to the LANA report was not available for this assessment. Kirti Joshi, in comments published two weeks after the report, called LANA's conclusion incorrect and argued that the project missed a requirement, the arithmetic Teichmüller datum of a fixed number field, that his own framework supplies together with a proof of the corollary; yet he also holds that Mochizuki's published procedures do not yield what the proof needs, so his dissent is about the diagnosis, not about whether the published text is complete. Whether the compatibility is the Scholze-Stix obstruction under another name is a separate question: LANA presents it as distinct, since the monodromy diagram of 2018 does not arise in its formulation. A proof of the compatibility accepted by the formalization team or by independent experts would show the published argument to have been incomplete rather than wrong and would call for this claim to be reread; a demonstration that the compatibility fails would sharpen it into a refutation of the step; a convincing showing that Mochizuki's intended derivation does not have the shape LANA reconstructed would undermine it.
Claim entered the graph