Kirti Joshi's arithmetic Teichmüller theory establishes the existence of distinct arithmetic holomorphic structures.
3 events · 1 assessment · 1 decision
Structured and assessed
First pass. Decomposed into three subclaims after match_claim found none existing: (1) plurality of untilts of a characteristic-p perfectoid field (requires, mathematics bedrock, importance 0.12, left as deferred stub), grouped under a "for" argument; (2) identity of Joshi's arithmetic holomorphic structures with Mochizuki's (defines, importance 0.4, contestation 0.75), grouped under an "against" argument, since the term's meaning is disputed and decides what the claim asserts; (3) validity of Joshi's proof of Corollary 3.12 (supports, importance 0.45). Written forms and evaluations recorded for both arguments. Recorded four instances: Joshi affirming in three arXiv documents (2022, 2024, 2025) and Mochizuki denying in his March 2024 report. Assessed CONTESTED at confidence 0.7 with no credence: the existence theorem is uncontroversial in Joshi's own sense but whether those are the structures at issue in the IUT dispute is denied by Mochizuki and doubted by IUT's critics, with no independent endorsement. Importance confirmed at 0.35 (contestation 0.6). Domain set to mathematics for future passes. Canonical form left unchanged: 14 words, neutral, affirmative, acceptable to both sides. Marginal yield 0.4: a mathematics-skilled pass could read Joshi's Construction I theorem statements and compatibility conditions directly rather than through the parties' summaries, and Scholze's 2024 comments were seen only via Woit's reporting. Five web searches used.
Assessed Contested
verdict confidence 0.70
Kirti Joshi's "Construction of Arithmetic Teichmüller Spaces" series (arXiv, 2021 to 2025) gives the phrase "arithmetic holomorphic structure", which Mochizuki introduced in inter-universal Teichmüller theory without a quantitative definition, a precise meaning: fix a variety over a p-adic field and a perfectoid field of characteristic p, and take the rigid analytic spaces obtained from the different untilts of that perfectoid field, which share a tempered fundamental group while differing in their arithmetic. In that sense the claim is essentially a theorem of standard p-adic geometry: a perfectoid field of characteristic p has infinitely many distinct untilts, parametrized by the Fargues–Fontaine curve, so Joshi's construction produces infinitely many distinct structures of his kind, and no one has disputed this as mathematics. What is disputed is whether these are the structures whose existence is at issue. The question arose because Mochizuki's proof of Corollary 3.12 compares and averages over distinct arithmetic holomorphic structures, and Scholze and Stix argued in 2018 that, as Mochizuki defines them, such structures cannot differ. Joshi holds that his definition is the canonical one, that his theory is the only one which establishes the structures' existence in the context of Mochizuki's papers, and that this refutes Scholze and Stix. Mochizuki's March 2024 report on Joshi's preprints replies that there are fundamental structural differences between Joshi's tilt-based construction and the constructions of IUT, and that Joshi's plurality of structures is never used in IUT's proofs. Commentators on the debate have made the complementary point that Scholze and Stix were speaking of Mochizuki's definitions, so that a theorem about differently defined objects cannot falsify their remark; Joshi himself has documented points of departure between the two theories. The claim therefore turns on whether Joshi's arithmetic holomorphic structures are the same objects as Mochizuki's, which both Mochizuki and the critics of IUT deny or doubt, and which no independent expert has endorsed. The strongest possible vindication would be acceptance of Joshi's proof of Corollary 3.12 within his framework, since that is the step where the distinct structures are actually compared. Scholze and Mochizuki both publicly criticized that proof in 2024, and although Joshi reports having fixed the issue raised and having received no further objections since October 2024, the work remains unrefereed and has not persuaded either side of the original dispute. A refereed validation of Joshi's series, or an explicit demonstration that his structures satisfy the compatibility conditions IUT requires of Mochizuki's, would resolve the question in his favour; a demonstration that the tilt-based objects cannot play the role of Mochizuki's structures would resolve it against him.
Claim entered the graph