Joshi's arithmetic holomorphic structures are the same objects as those of Mochizuki's inter-universal Teichmüller theory.
Assessment
Available evidence weighs against the claim.
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.
Full reasoning: the evidence and decisions behind this verdict
Sources read whole: Mochizuki's March 2024 "Report on the recent series of preprints by K. Joshi" (www.kurims.kyoto-u.ac.jp/~motizuki/Report%20on%20a%20certain%20series%20of%20preprints%20(2024-03).pdf); Joshi's "Comments on Arithmetic Teichmuller Spaces" (arxiv.org/pdf/2111.06771); Joshi's "Final Report on the Mochizuki-Scholze-Stix Controversy" (arxiv.org/pdf/2505.10568); the abstract of Joshi's "Untilts of fundamental groups" (arxiv.org/abs/2210.11635); Scholze's answer and comments on MathOverflow (mathoverflow.net/a/435112); and the comment thread on Woit's July 2023 post (www.math.columbia.edu/~woit/wordpress/?p=13573), which carries both Sawin's denial and Joshi's reply.
Instances. Affirming: Joshi, three documents (2022 abstract: his structures "also provide" Mochizuki's; 2022 Comments: "arithmetic holomorphic structures in the literal sense(!)... many distinct labels in Mochizuki's sense too"; 2025 Final Report: "a precise definition", "a canonical definition", "one and only one theory which can be constructed using Mochizuki's Key Principle"), plus his July 2023 blog reply that his definition "provides arithmetic holomorphic structures in Mochizuki's sense" with "no change in the essential content" (same page as Sawin's comment, so not separately recorded). Denying: Scholze (November 2022), Sawin (July 2023), Mochizuki (March 2024). Every affirmation is the author's own; no independent voice affirms. Two recorded passages from PDFs failed the mechanical quote check only because the stored text renders "Teichmüller" and dashes with extraction artifacts; both passages were read in the source.
Why contradicted rather than contested. First, the textual facts are not in dispute: the IUT papers contain no perfectoid fields, tilts or untilts, which Joshi's Final Report concedes ("there is no mention of the required input data in [Mochizuki]"); the objects Joshi constructs are Berkovich spaces over distinct untilts, non-isomorphic as such, whereas Mochizuki's 2024 report says IUT's plurality holds "relative to a compatibility condition" and is "an immediate consequence of the definition of a ring". Second, Joshi's own account undercuts literal identity: the Comments paper calls his approach "fundamentally different", says IUT's indexing set is arbitrary while his is fixed by pairs giving rise to the same tempered fundamental group data, and describes Mochizuki's indeterminacies as "corresponding" to his; the Final Report calls IUT a "special case" of his theory. Third, Scholze's objection is specific and survives scrutiny: in Joshi's Theorem 4.8 the curve and its tempered fundamental group are fixed and only the embedding of the base field into an untilt varies, so this is data the anabelian side cannot see; Joshi's answer is that this is precisely the Teichmüller freedom, which concedes the point about the data and shifts the claim to what IUT ought to have used. Fourth, the author of the term denies the identification with reasons, and the expert in the relevant machinery denies it with reasons, while the affirmation rests on Joshi's reading of the Key Principle, which Mochizuki rejects and for which no argument is given that other base-point data could not serve.
Why not more confident. Mochizuki's IUT papers give no definition of arithmetic holomorphic structure precise enough to prove that two such structures differ is plausibly true (Dupuy and others have noted the ambiguity), and if the target notion is underspecified, "same objects" is partly an interpretive judgment; a reader who accepts Joshi's reading of the Key Principle could hold the identification as a charitable reconstruction. Mochizuki's report states the structural differences only by citing items in his earlier "Essential Logical Structure" report, which was not opened here, and he admits his understanding of tilts is limited. The residual credence of about 0.15 reflects that the literal identity is very unlikely but that the notion's imprecision leaves room for the reconstruction reading.
How the subclaims weigh. The untilt data that defines Joshi's structures plays no role in the constructions of the IUT papers is the decisive premise and is close to a textual fact. In IUT, distinct arithmetic holomorphic structures means ring structures related by a non-ring-theoretic link is Mochizuki's own account and, if accepted, makes the two notions different in kind. The Key Principle requires arbitrary geometric base-points that only perfectoid fields supply is the hinge of Joshi's case and is asserted without a supporting argument for its exclusivity conjunct. Joshi's theory yields a valid proof of Corollary 3.12 would be strong indirect support if it held with Mochizuki's content; it is unrefereed, criticized by Scholze (purely local proof) and Mochizuki (local inequalities cannot sum to a global one), and so adds little now.
What would change the verdict: an independent expert reading of Construction I to IV confirming that Joshi's structures reproduce IUT's Hodge theaters and Θ-link with the same content (toward contested or supported); a refereed acceptance of Joshi's Corollary 3.12 with Mochizuki's global content (toward contested); or, in the other direction, a written account of Mochizuki's cited structural differences (TltDf1 to TltDf7) that an independent reader endorses (toward higher confidence in contradicted).
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.
Because Mochizuki's IUT papers never define arithmetic holomorphic structure precisely enough to prove that two such structures differ, the notion IUT relies on is fixed only by the role it must play; and because Mochizuki's Key Principle of Inter-Universality demands arbitrary geometric base-points that only algebraically closed perfectoid fields supply, structures indexed by untilts of such fields are the only precise notion that can fill that role, so Joshi's structures are the structures IUT was implicitly about. That Joshi's theory yields a proof of Mochizuki's Corollary 3.12 would corroborate the identification by showing his structures doing the work IUT assigns to its own.
Granting its premises the inference reaches only a reconstruction: if IUT's notion is fixed by its role and only untilt-indexed structures can fill that role, Joshi's structures are the best available precisification, which is weaker than their being the same objects. The argument lives or dies on the reading of the Key Principle as demanding base-points that only perfectoid fields supply, which Mochizuki rejects and which is asserted without an argument for its exclusivity; the imprecision of Mochizuki's definition is plausible but cuts both ways, since an underspecified target cannot be shown identical to anything. Joshi's claimed proof of Corollary 3.12 would corroborate the argument only if it carried Mochizuki's global content, which Scholze and Mochizuki dispute.
Joshi's structures are indexed by untilts of a characteristic-p perfectoid field and are non-isomorphic as Berkovich spaces, whereas in IUT "distinct arithmetic holomorphic structures" means ring structures related by a non-ring-theoretic link, not non-isomorphic structures; and since the untilt data defining Joshi's structures plays no role in the constructions of the IUT papers, the extra data Joshi calls an arithmetic holomorphic structure is foreign to IUT, so the two notions share a name but not an identity.
The inference goes through: if the data indexing Joshi's structures does no work in IUT and IUT's plurality is a matter of ring structures related by a link rather than of non-isomorphic objects, then Joshi's non-isomorphic Berkovich spaces are a different notion under the same name. The argument rests chiefly on the untilt data playing no role in IUT's constructions, which is close to a textual fact about the published papers and which Joshi's own remark that IUT names no such input concedes; Mochizuki's account of what distinctness means in IUT is the author's own statement and adds a difference in kind. Joshi's reply, that IUT ought to have used this data, does not touch either premise.
Provenance
Where this claim has been said, linked to its canonical form.
Every affirmation of the identification comes from Kirti Joshi himself, across three of his own documents from 2022 to 2025, and his earliest comparison paper concedes that the two approaches are fundamentally different in method even as it asserts the structures coincide. The denials come from three independent voices: Peter Scholze, replying directly to Joshi's 2022 preprint with a specific argument about what its main theorem varies; Will Sawin, arguing from Joshi's own acknowledged differences; and Mochizuki, whose report states the differences by reference to his earlier writing and admits limited understanding of tilts. No source independent of the parties affirms the claim, so the instance count reflects one author against three critics, not a balanced literature. A reader should open Scholze's answer first and Joshi's Comments paper second.
Arithmetic holomorphic structures introduced here also provide distinct arithmetic holomorphic structures used by Shinichi Mochizuki in [Mochizuki,2021a,b,c,d].
Abstract of Joshi's announcement paper, asserting that the arithmetic holomorphic structures he constructs via untilts of perfectoid fields are the ones Mochizuki's IUT papers use, and framed as answering the Scholze–Stix question of whether distinct such structures exist in IUT.
Asserted without evidence of the source's own. Only the abstract page was read; the abstract states the identification as a consequence of the paper without argument. Scholze's reply on MathOverflow addresses this paper's Theorem 4.8 directly. Worth reading closely: Definition 5.1 and the discussion of Scholze–Stix in this paper are where Joshi's definition is stated and where he argues it provides structures in Mochizuki's sense; a close reading would show how much of the identification is argued rather than asserted.
Joshi defines a category $C'$ with a non-fully-faithful forgetful functor $C'\to C$ by endowing objects of $C$ with some extra data, and then notes that $C'\to C\to D$ is not fully faithful. Joshi makes the linguistic trick of calling the extra data he puts on objects of $C$ an "arithmetic holomorphic structure", but this is just linguistics...
Scholze's answer and follow-up comment on Joshi's 2022 "Untilts" preprint, holding that Joshi's Theorem 4.8 does not falsify Remark 9 of Scholze–Stix because the extra untilt data Joshi adds is unrelated to the tempered fundamental group and that calling it an "arithmetic holomorphic structure" is a change of terminology rather than the notion at issue.
The source's own evidence bears what it asserts. Scholze gives a specific argument: in Joshi's Theorem 4.8 the curve and its tempered fundamental group stay fixed and only the embedding of the base field into an untilt varies, so the added data is not something the fundamental group could see, and calling it an arithmetic holomorphic structure is a relabeling. The argument is brief but self-contained and addresses the definition rather than the reputation of the work. Worth reading closely: It is the most precise independent expert statement of why Joshi's notion is taken to differ from Mochizuki's, and Joshi's later writings respond to it.
The definition of "arithmetic holomorphic structures" you use in your work is – as you point out in the linked Quest paper, for example – different from Mochizuki's. Surely Scholze and Stix were thinking about Mochizuki's definitions when they made their claim that distinct arithmetic holomorphic structures do not exist. The fact that, under your definition, they do exist, is not a falsification.
Will Sawin, replying to Joshi in the comment thread, arguing that Joshi's notion of arithmetic holomorphic structure is a different definition from Mochizuki's, so exhibiting many structures in Joshi's sense does not falsify the Scholze–Stix claim about Mochizuki's.
Asserted without evidence of the source's own. Sawin's point is logical rather than technical: he cites Joshi's own Quest paper as acknowledging that the definitions differ, and infers that existence under Joshi's definition does not touch the Scholze–Stix claim about Mochizuki's. Joshi replies in the same thread that his definition provides structures in Mochizuki's sense with no change in essential content, which is the affirmation of the claim by its author in the same source.
Moreover, by considering tilts, one can indeed construct a situation [cf. [EssLgc], Example 3.5.3, (vi), (RCRS3), (RCΘ3)] that is roughly reminiscent of the situation surrounding the domain and codomain of the Θ-link in inter-universal Teichmüller theory [cf. [EssLgc], Example 3.5.3, (iv), (TltSim)], and which, moreover, satisfies the non-isomorphicity condition of (Js1-1), but which is, however, completely useless from the point of constructing a theory that is structurally similar to inter-universal Teichmüller theory, on account of the numerous and quite fundamental structural differences between this tilt-based construction and the corresponding constructions in inter-universal Teichmüller theory
Mochizuki's response to Joshi's Construction III and IV preprints, distinguishing the tilt-based plurality of non-isomorphic structures from the plurality of arithmetic holomorphic structures in IUT (which he says holds relative to a compatibility condition and is an immediate consequence of the definition of a ring) and listing fundamental structural differences between the two.
Asserted without evidence of the source's own. The report states the structural differences between the tilt-based construction and IUT by pointing to a list of items in the author's earlier "Essential Logical Structure" report rather than arguing them here; it also concedes that the author's understanding of tilts and untilts is limited and superficial. The report's own account of what plurality means in IUT (a compatibility condition on the Θ-link, an immediate consequence of the definition of a ring) is the substantive content bearing on the claim. Worth reading closely: It is the only statement from IUT's author of what "distinct arithmetic holomorphic structures" means in IUT and why the tilt-based notion is said to differ; the referenced items (TltDf1) to (TltDf7) in the Essential Logical Structure report would give the actual differences. The quoted passage was not found in the stored copy of this source.
My work ([Joshi, 2021a], [Joshi, 2022]) provides a precise definition of 'Arithmetic Holomorphic Structures' and this allows me to (1) show that these arise from rigid analytic spaces i.e. arise from p-adic holomorphic (or analytic) functions–hence one has the p-adic analog of classical Teichmüller Theory; (2) explicitly exhibit the existence of many such structures and, (3) allows one to elaborate and transparently prove the properties (of such structures) which Mochizuki has claimed in his work and requires in his proofs.
Joshi's 2025 report, holding that Mochizuki's own quantification of "arithmetic holomorphic structure" is inadequate, that his definition is the canonical one, and that his is the only theory consistent with Mochizuki's Key Principle of Inter-Universality; he also describes his theory as including IUT as a special case.
Asserted without evidence of the source's own. The report is a tabulation of the author's conclusions with pointers to his own preprints for proofs; it argues that Mochizuki's notion was never quantified and that a canonical definition exists, but the identification itself rests on the author's reading of Mochizuki's Key Principle rather than on anything demonstrated in the document. It also describes IUT as a special case of the author's theory, which is a weaker relation than identity. The quoted passage was not found in the stored copy of this source.
In [Joshi, 2021a], I substantially deepen this idea: in my theory labels correspond to distinct (arithmetic) Berkovich analytic (i.e. holomorphic) structures. 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 early comparison of his theory with IUT. He asserts that his Berkovich-analytic structures realize Mochizuki's labels "in the literal sense", while in the same document describing his approach as "fundamentally different" from Mochizuki's, listing several points of departure (local-to-adelic order, use of perfectoid fields instead of anabelian reconstruction, indexing set), and saying Mochizuki's indexing set is arbitrary where his is constrained.
The assertion outruns the source's own evidence. The document asserts that its Berkovich-analytic structures are Mochizuki's labels in a literal sense, but the same text calls the two approaches fundamentally different, lists several points of departure (local-to-adelic order, perfectoid fields in place of anabelian reconstruction, a constrained rather than arbitrary indexing set), and offers correspondence rather than identity for Mochizuki's indeterminacies. Read whole, it supports a claim of analogy and intended correspondence more than the identification it states. Worth reading closely: It is Joshi's own most candid account of how the two theories differ, and the differences he lists are the material the denials rely on. The quoted passage was not found in the stored copy of this source.
How these sources relate
- https://mathoverflow.net/a/435112 responds to https://arxiv.org/abs/2210.11635, faithfully. Scholze's answer is a direct rebuttal of the identification asserted in the abstract of Joshi's 2022 Untilts preprint, engaging its Theorem 4.8 specifically. His description of what varies in that theorem (the embedding into an untilt, with the curve fixed) matches Joshi's own description of his construction in the Comments paper.
- https://www.math.columbia.edu/~woit/wordpress/?p=13573 cites in support Mochizuki's Corollary 3.12 and my quest for its proof (Kirti Joshi). Sawin offers Joshi's own Quest paper as acknowledging that the two definitions differ. The Quest paper was not opened in this pass, so whether it concedes a difference in definition or only a difference in method cannot be confirmed; Joshi's Comments paper, which was read, does describe the approaches as fundamentally different while insisting the structures coincide in essential content. Judged from the citing document alone.
- 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).
- https://arxiv.org/abs/2210.11635 and https://arxiv.org/pdf/2505.10568 share an author. Both are single-author documents by Kirti Joshi (arXiv bylines).
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