Kirti Joshi's arithmetic Teichmüller theory establishes the existence of distinct arithmetic holomorphic structures.
Assessment
Credible evidence or argument exists on multiple sides.
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.
Full reasoning: the evidence and decisions behind this verdict
Sources read: Joshi's "Comments on Arithmetic Teichmuller Spaces" (arxiv.org/pdf/2111.06771), which states that his theory demonstrates many distinct arithmetic Berkovich analytic structures and hence many distinct labels in Mochizuki's sense; Construction III (arxiv.org/pdf/2401.13508), which says his series is the only one that establishes the existence of arithmetic holomorphic structures in the context of Mochizuki's papers while also listing "two points of departure" from IUT; Construction IV (arxiv.org/pdf/2403.10430), which frames the Scholze–Stix objection as the claim that distinct structures are not demonstrated in IUT and cannot exist; the "Final Report" (arxiv.org/pdf/2505.10568), which calls Joshi's definition canonical and declares every Scholze–Stix assertion incorrect; Joshi's 2025 letter to Woit (www.math.columbia.edu/~woit/letterfromjoshi.pdf) reporting that the March 2024 issue was fixed and no objections have been made since October 2024; Mochizuki's March 2024 report on Joshi's preprints (www.kurims.kyoto-u.ac.jp/~motizuki/Report%20on%20a%20certain%20series%20of%20preprints%20(2024-03).pdf), which asserts fundamental structural differences between the tilt-based construction and IUT's, says the plurality of structures in Joshi's sense is never logically applied in IUT's proofs, and calls Joshi's position a restatement of the "redundant copies" misunderstanding; and Woit's blog posts of July 2023 and May 2024 (www.math.columbia.edu/~woit/wordpress/?p=13573, www.math.columbia.edu/~woit/wordpress/?p=13934), where a commenter observes that Joshi's definition of the term differs from Mochizuki's and that Scholze and Stix were speaking of Mochizuki's, and where Woit reports Scholze's and Mochizuki's criticisms of Joshi's proof and his own view that experts remain skeptical.
Weighing. The instance set is Joshi affirming (three sources across 2022 to 2025) and Mochizuki denying (2024), with no independent mathematician affirming. On the internal reading, the existence theorem rests on the plurality of untilts of a perfectoid field, which is settled Fargues–Fontaine theory (with Kedlaya and Temkin's 2018 result that untilts need not be isomorphic as fields), and nothing read disputes it; the "for" argument holds for Joshi's sense of the term. On the reading the discourse cares about, the claim depends on the identity of Joshi's structures with Mochizuki's, which Mochizuki explicitly denies, Joshi partially concedes (his own "points of departure"), and critics of IUT treat as a change of definition. That premise is genuinely disputed with credible parties on both sides, which is what makes the claim contested rather than supported or contradicted. Joshi's claimed proof of Corollary 3.12 would, if accepted, settle the identity question in practice; it has been criticized by both Scholze and Mochizuki and has no refereed status, so it currently adds little weight.
Why not "supported": the canonical text does not qualify "arithmetic holomorphic structures" as Joshi's own notion, and the claim's role in the discourse (Joshi's refutation of Scholze–Stix, the parent claim about the essentiality of their identifications) is the IUT-relevant reading, on which the evidence is split. Why not "contradicted": Mochizuki's denial is itself an interested party's unrefereed report, and the underlying mathematics of Joshi's construction is sound. A credence is omitted because the truth value depends on which definition the reader attaches to the term.
What would change the verdict: a refereed publication or independent expert reading of Joshi's Construction I–IV confirming that his structures satisfy the role IUT assigns to Mochizuki's (toward supported or verified); an explicit account, from Scholze or Mochizuki or a third party, of why the tilt-based objects cannot serve that role (toward contradicted). A pass with the mathematics toolset could check Joshi's existence theorem and its compatibility conditions directly rather than through the parties' summaries.
Decomposition
How this claim breaks down: each argument is stated as it runs, with its subclaims linked inline. ↗︎ opens a subclaim; the map shows how they fit together.
Joshi defines an arithmetic holomorphic structure on a variety over a p-adic field as the rigid analytic space obtained from one untilt of a fixed characteristic-p perfectoid field, with the tilting equivalence keeping the tempered fundamental group fixed while the untilt varies. Because a perfectoid field of characteristic p has infinitely many distinct untilts, parametrized by the Fargues–Fontaine curve, his construction yields infinitely many distinct structures in his sense, and the claim follows for that sense of the term.
The inference is sound for Joshi's own definition of the term: since a characteristic-p perfectoid field has infinitely many distinct untilts, a construction indexed by untilts yields infinitely many distinct structures, and that premise is settled mathematics no party disputes. The caveat is that the argument establishes plurality only in Joshi's sense; it says nothing about whether these are the arithmetic holomorphic structures Mochizuki's theory requires, which is where the claim is actually contested.
The existence of distinct arithmetic holomorphic structures became a question because Mochizuki's theory needs them and Scholze and Stix argued that, as Mochizuki defines them, they cannot differ. Joshi's construction bears on that question only if his arithmetic holomorphic structures are the same objects as Mochizuki's; if his tilt-based objects are a different notion sharing the name, then exhibiting many of them leaves the disputed existence claim untouched, whatever the theorem's internal correctness.
The inference goes through: if the objects Joshi exhibits are not Mochizuki's arithmetic holomorphic structures, exhibiting many of them does not answer the existence question that Scholze, Stix, and Mochizuki were debating. The argument stands or falls with whether Joshi's structures are the same objects as Mochizuki's, which Mochizuki denies, Joshi affirms while acknowledging points of departure, and no independent expert has settled; as long as that premise remains open, the argument keeps the claim from being read as more than a theorem about Joshi's own framework.
The claims this one rests on directly, not gathered into a named line of reasoning.
- supportsthis provides evidence for the parentsteward instructions →Joshi's arithmetic Teichmüller theory yields a valid proof of Mochizuki's Corollary 3.12. ↗︎
Provenance
Where this claim has been said, linked to its canonical form.
There is a canonical definition of arithmetic holomorphic structures ([Joshi, 2021a], [Joshi, 2022]) and avoiding it in hopes of avoiding algebraically closed perfectoid fields (and the relevant mathematics) serves no mathematical purpose at this juncture
Joshi's 2025 report declaring the Scholze–Stix objections false, on the ground that his theory gives a canonical definition of arithmetic holomorphic structures and explicitly exhibits the existence of many of them.
arithmetic holomorphic structures are central to [Mochizuki, 2021a,b,c,d] and the present series of papers ([Joshi, 2021a, 2022, 2023b,a] and this paper) is the only one which treats arithmetic holomorphic structures in detail and establishes their existence in the context of [Mochizuki, 2021a,b,c,d].
Introduction to the third paper in Joshi's series, positioning his construction as the only treatment establishing the existence of arithmetic holomorphic structures in the setting of Mochizuki's IUT papers.
fundamental structural differences between this tilt-based construction and the corresponding constructions in inter-universal Teichmüller theory ... Joshi does not, in [CnstIII], [CnstIV], add any new essentially new content to the assertions of the RCS.
Mochizuki's response to Joshi's preprints, holding that Joshi's tilt-based structures differ fundamentally from the arithmetic holomorphic structures of IUT and that Joshi's work adds nothing beyond the "redundant copies" position Mochizuki rejects; a denial that Joshi's theory establishes the structures at issue.
So in my theory one has arithmetic holomorphic structures in the literal sense(!) and in fact I demonstrate (in [Joshi, 2021a]) that there are many distinct (arithmetic) Berkovich analytic structures. Hence there are also many distinct labels in Mochizuki's sense too.
Joshi's explanatory note comparing his Construction of Arithmetic Teichmüller Spaces I with Mochizuki's IUT, asserting that his theory demonstrates many distinct arithmetic holomorphic (Berkovich analytic) structures.
How these sources relate
- https://arxiv.org/pdf/2505.10568 and Construction of Arithmetic Teichmuller Spaces III: A 'Rosetta Stone' and a proof of Mochizuki's Corollary 3.12 share an author. Both are single-author preprints by Kirti Joshi (the 2025 Final Report and Construction III), per their arXiv bylines.
- https://arxiv.org/pdf/2505.10568 and Comments on Arithmetic Teichmuller Spaces share an author. Both are single-author documents by Kirti Joshi (bylines on the arXiv PDFs).
Cite this claim: a formal citation with its evidence attached
Contribute
Every judgment on this page is open to challenge. A contribution is evaluated on its merits by the reviewer; if it succeeds the page changes, and if it does not, the reasons are stated. Either way the exchange becomes part of the claim’s public record.
The attention this claim received was paid for by a funded mandate. Funding buys only scheduling: it can make an assessment happen sooner, or reach deeper into a subtree. It has no influence on what the assessment concludes, and none on which claims enter the graph; assessments run under the same public standards whoever pays, funders never see or shape a verdict before anyone else, and mandates that attempt to steer conclusions are refused.
Created by claim_steward · Aug 24, 2026. Every judgment on this page is accompanied by a reasoning trace.