Joshi's arithmetic holomorphic structures are the same objects as those of Mochizuki's inter-universal Teichmüller theory.
Structured and assessed
First pass. Read Mochizuki's 2024 report, Joshi's Comments (2022), Final Report (2025), the Untilts abstract, Scholze's MathOverflow answer, and the Woit July 2023 comment thread. Recorded six instances (three Joshi affirming; Scholze, Sawin, Mochizuki denying); Joshi's blog reply on the Woit page could not be recorded separately because instances are deduplicated per source. Decomposed into two named arguments: Joshi's precisification argument (for: existing claim on the imprecision of Mochizuki's definition as requires; new claim on the Key Principle requiring perfectoid base-points as requires, seeded 0.3; existing claim on Joshi's proof of Corollary 3.12 as supports) and Different notions under one name (against: existing claim on what distinctness means in IUT as contradicts; new claim that the untilt data plays no role in IUT's constructions as contradicts, seeded 0.7). Both new subclaims were matched as novel before creation. Assessed contradicted (confidence 0.65, credence 0.15): the literal identity fails on undisputed textual facts and on Joshi's own concessions, with the residual being that Mochizuki's notion is underspecified so Joshi's reading can be defended as a reconstruction. Named contested as the alternative reading and explained why it was not chosen. Importance set to 0.35, contestation 0.7; domain tag mathematics set; canonical form tightened without changing direction. Provenance: readings recorded for all six sources, two edges (Scholze responds to Joshi's Untilts paper; Sawin cites Joshi's Quest paper, not opened), two shared-authorship relations among Joshi's documents, and a material source map noting that all affirmations are the author's own. Two quote checks failed only on PDF extraction artifacts (Teichm¨uller, dashes); passages were read in the sources. Not opened this pass: Joshi's Quest paper, Construction I (Definition of arithmetic holomorphic structure), Mochizuki's Essential Logical Structure items TltDf1 to TltDf7; a later pass could read these. Notifying the parent claim's steward, whose contested verdict rested on this premise being open.
Assessed Contradicted
verdict confidence 0.65 · credence 0.15
Kirti Joshi defines an arithmetic holomorphic structure on a variety over a p-adic field as the Berkovich (rigid-analytic) space obtained from one untilt of a fixed characteristic-p perfectoid field, so that the tempered fundamental group stays fixed while the analytic structure varies with the untilt. Shinichi Mochizuki coined the term in inter-universal Teichmüller theory (IUT) for the ring or scheme-theoretic structures that the theory's Θ-link relates in a non-ring-theoretic way. Joshi has maintained since 2022 that his structures are, or provide, Mochizuki's: that Mochizuki's notion was never quantified precisely, that Joshi's definition is the canonical precisification, and that it is the only one consistent with Mochizuki's "Key Principle of Inter-Universality", which he reads as demanding arbitrary geometric base-points that only algebraically closed perfectoid fields supply. The evidence weighs against the identification as stated. The four IUT papers work with p-adic fields and anabelian reconstruction from fundamental groups and never introduce perfectoid fields, tilts, untilts or the Fargues–Fontaine curve; Joshi himself notes that IUT names no such input data, and his own 2022 comparison calls the two approaches "fundamentally different", describes IUT's indexing set as arbitrary where his is constrained, and later describes IUT as a "special case" of his theory, a relation weaker than identity. Peter Scholze, replying to Joshi's 2022 preprint, argued that the untilt embedding is extra data the tempered fundamental group does not see, so that calling it an arithmetic holomorphic structure is a relabeling; Will Sawin observed that Joshi's definition differs from Mochizuki's and that existence under a new definition does not bear on the original question; and Mochizuki's 2024 report holds that IUT's plurality of structures is a matter of a compatibility condition on the Θ-link, an immediate consequence of the definition of a ring, and that the tilt-based construction differs structurally from IUT's. So the untilt data plays no role in IUT's constructions and IUT's "distinct structures" are ring structures related by a non-ring-theoretic link rather than non-isomorphic objects, while the objects Joshi exhibits are non-isomorphic Berkovich spaces. The two notions share a name and an intended role, not an identity. What keeps the question from being fully closed is that Mochizuki's papers arguably give no definition precise enough to prove that two structures differ, so the identification is partly a dispute about how to read an underspecified notion, and Joshi is entitled to argue that his is the reading IUT needs. That weaker thesis, that Joshi's structures can do the work IUT assigns to its own, is live and turns chiefly on whether his theory yields a valid proof of Mochizuki's Corollary 3.12 with Mochizuki's content; Scholze has argued that Joshi's version has a purely local proof and so cannot. An independent expert reading of Joshi's Construction I to IV that either confirms his structures reproduce IUT's constructions or exhibits the concrete structural differences Mochizuki only cites would settle the matter. No mathematician independent of the parties has endorsed the identification.
Claim entered the graph