Mochizuki's IUT papers establish the existence of distinct arithmetic holomorphic structures.
Assessment
Available evidence weighs against the claim.
The question is whether Mochizuki's four inter-universal Teichmüller theory papers prove, rather than posit, that there are two or more genuinely different arithmetic holomorphic structures, the objects whose comparison is supposed to yield the inequality of Corollary 3.12. The answer depends on what "distinct" is taken to mean, and the discourse has settled on the stronger meaning.
In the papers' own usage, an alien arithmetic holomorphic structure is simply the ring structure of a Hodge theater related to a given one by a link that is not ring-theoretic, and since the Θ-link is such a link, the papers do exhibit distinct structures in that relational sense. Mochizuki's 2018 report states the point this way: the holomorphic structures on the two sides of the Θ-link must be treated as distinct because they are related in a way that is not ring-theoretic. No one disputes the premises of this argument; the dispute is whether its conclusion amounts to anything.
Every examiner outside Mochizuki's circle who has addressed the point says it does not. The Hodge theaters on the two sides of the Θ-link are isomorphic copies of one structure, and Scholze and Stix argued in 2018 that identifying them along an isomorphism loses nothing and makes the key theorem trivial. Kirti Joshi, who unlike Scholze and Stix believes that distinct arithmetic holomorphic structures exist and has built a theory to construct them, agrees with them on this narrower point: the IUT papers give no definition under which two such structures can be shown unequal, so their existence is suggested there but not established. The LANA formalization project's 2026 interim report, working sympathetically within Mochizuki's framework, isolated an unproven compatibility at exactly the stage where the structures are compared and declined to give a verdict.
The claim is therefore best read as contradicted in the sense in which it is debated, with the qualification that it is true, and trivially so, under Mochizuki's relational definition. What would change the verdict is a demonstration, accepted outside Mochizuki's circle, that the structures related by the Θ-link differ in some isomorphism-invariant respect the proof actually uses, or an independent formalization that reconstructs the comparison across them.
Full reasoning: the evidence and decisions behind this verdict
Sources. Mochizuki's Report on Discussions (2018, www.kurims.kyoto-u.ac.jp/~motizuki/Rpt2018.pdf) states that in IUT one must treat the holomorphic structures in the domain and codomain of the Θ-link as distinct structures related in a nontrivial way; the introduction to IUT IV (www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf) glosses an alien arithmetic holomorphic structure as the ring/scheme structure of a Hodge theater related to a given one by a non-ring/scheme-theoretic horizontal arrow of the log-theta-lattice. Joshi's Construction III (arxiv.org/pdf/2401.13508, section 1.12) says the existence of distinct arithmetic holomorphic structures is suggested in Mochizuki's 2022 writings but not clearly established in the IUT papers and cannot be quantitatively established in their framework; his Final Report (arxiv.org/pdf/2505.10568, section 1.2) says Mochizuki's quantification of the notion is inadequate to assert that one has two or more such structures, and its Table 2 records that Mochizuki asserts the existence of distinct Hodge theaters, étale pictures and Frobenius pictures without proof; his November 2025 FAQ says he agrees with Scholze and Stix that the papers lack the language to assert that two structures are unequal. Joshi's Construction IV (arxiv.org/pdf/2403.10430, section 1.9) summarizes Scholze and Stix's first core objection as being that distinct arithmetic holomorphic structures are not demonstrated in the IUT papers and cannot exist given Mochizuki's absolute anabelian reconstruction theorem. Scholze and Stix's own report (ncatlab.org/nlab/files/why_abc_is_still_a_conjecture.pdf) denies the claim in substance by arguing that identifying the copies along the identity makes Theorem 3.11 trivial rather than false. The LANA interim report (ncatlab.org/nlab/files/LANAProject-Report-July2026.pdf) speaks of the q-pilot "in its native arithmetic holomorphic structure" and reports that the compatibility between that construction and the multiradial one has not been reconstructed, without offering a verdict. Joshi's July 2026 comments on the LANA report restate his position that intrinsic labels are needed to assert the plurality Mochizuki relies on. None of the PDF sources could be opened whole on this pass because the reading tool failed on them; the passages above were read as excerpts and the exact wording of the Scholze–Stix report was not re-checked here.
Instances. One affirmation, Mochizuki's 2018 report. Three denials, all Joshi's (2024, 2025, 2025), so the denials are one voice restated. Scholze and Stix's denial is recorded on the parent claim rather than here because their report does not use the phrase. There is no affirmation from anyone outside Mochizuki's circle; Hoshi's and Fesenko's expository writings were not read on this pass and would be worth recording if they assert the point in their own voice.
How the subclaims weigh. The two premises of the affirming line are both essentially uncontested: the Θ-link is not ring-compatible, and the papers do use "distinct arithmetic holomorphic structure" relationally. Granting them yields the claim only under that relational usage, on which "distinct" adds nothing to "related by the Θ-link". The claim as debated is the stronger one, because what Corollary 3.12 needs is a plurality of structures whose arithmetic quantities can be compared and averaged, and that is the sense in which Joshi denies it and Scholze and Stix deny it. On that reading the two contradicting premises decide the matter: the Hodge theaters on both sides are isomorphic (uncontested), and the papers supply no definition under which inequality of two structures can be proved (Joshi's position, denied by Mochizuki's 2024 report on Joshi's preprints, but consistent with the papers' treatment of the term as a gloss on ring structure rather than as a defined object with an equality relation).
Why contradicted rather than contested. The affirming party is the author, and his affirmation, read closely, is a stipulation about how the theory must be read rather than a proof that two structures differ. The denying parties include both the theory's principal critics and its most sympathetic outside reconstructor, who otherwise disagree about almost everything. The one 2026 development that might have moved the claim, the LANA report, stalled at precisely this point. The verdict is held at moderate confidence because the claim's truth turns on a definitional choice that the canonical wording does not fix, and a reader who adopts Mochizuki's usage will reasonably regard it as true; contested is the alternative status, and would become the right one if an independent party affirmed the substantive reading.
What would change the verdict. A proof, accepted outside Mochizuki's circle, that the structures on either side of the Θ-link differ in an isomorphism-invariant respect the proof uses (toward supported or verified); a LANA formalization that reconstructs the compatibility between the native and multiradial constructions (toward supported); or a refereed acceptance of Joshi's constructions together with a demonstration that they coincide with Mochizuki's structures, which would establish existence but by Joshi's theory rather than by the IUT papers and so would leave this claim where it is. A full reading of Mochizuki's 2018 report, sections (GLR2) and (GIUT), and of the LANA report's section 10 would sharpen the confidence in either direction.
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.
IUT's own gloss is that an alien arithmetic holomorphic structure is a ring structure related to a given one by a link that is not ring-theoretic, and the Θ-link is exactly such a link, so on that usage the domain and codomain of the Θ-link are distinct arithmetic holomorphic structures and the papers exhibit them by constructing the Θ-link.
Both premises are common ground: the Θ-link is not compatible with ring structures, and the papers use "distinct arithmetic holomorphic structure" relationally. The inference goes through, but only to the conclusion that the two sides of the Θ-link are distinct in the papers' own relational sense, which is a matter of definition rather than proof. It does not reach the sense in which the claim is debated, that two structures exist whose difference the proof of Corollary 3.12 can exploit, so the argument establishes the claim only on a reading its critics regard as empty.
Because the Hodge theaters on the two sides of the Θ-link are isomorphic, any two structures the papers call distinct are copies of one structure, and since the papers give no definition under which two arithmetic holomorphic structures can be proved unequal, the distinctness is asserted by labeling rather than established by proof.
The inference is valid: isomorphic copies with no definable inequality between them cannot have been proved distinct in any sense stronger than labeling. Its first premise, that the Hodge theaters on the two sides of the Θ-link are isomorphic, is not seriously disputed. The argument's weight rests on the absence in the IUT papers of a definition under which two arithmetic holomorphic structures can be proved unequal, which Joshi asserts and Mochizuki denies; the papers' treatment of the term as a gloss on ring structure rather than as a defined object favors Joshi, but no third party has adjudicated it.
Provenance
Where this claim has been said, linked to its canonical form.
The affirmation comes from Mochizuki alone, in his 2018 report on the Kyoto discussions, and consists in the statement that the two sides of the Θ-link must be treated as distinct because the link is not ring-theoretic. The three denials are one voice, Kirti Joshi's, restated across a 2024 preprint, a 2025 report, and a 2025 FAQ; Joshi denies the claim in order to position his own theory as the construction that supplies what Mochizuki's papers lack. Scholze and Stix's 2018 report denies the claim in substance but does not use the phrase, and none of these documents could be opened whole on this pass; a reader should begin with Mochizuki's report and Joshi's 2025 report.
At the very center of the issue is that Mochizuki's quantification of what it means to be an Arithmetic Holomorphic Structure is mathematically inadequate to quantitatively assert that one has two or more such structures.
Joshi's section on why Mochizuki's proof is incomplete: Mochizuki correctly surmised that a Teichmüller theory of number fields exists but his anabelian methods cannot demonstrate the plurality of arithmetic holomorphic structures the proof needs to compare and average over.
Joshi's denial is a judgment about the adequacy of Mochizuki's definitions, offered by the author of a rival construction; the report argues the point by reference to his own earlier papers rather than by a passage-level analysis of where the IUT papers fall short. Only excerpts could be read on this pass. Worth reading closely: Section 1.2 and Table 2 are where Joshi lays out exactly which assertions of Mochizuki he considers unproven and which of Scholze and Stix he considers false; a close reading would show whether his denial of this claim rests on more than the absence of an equality relation in IUT's notion.
Notably, [Mochizuki, 2022] suggests the existence of distinct arithmetic holomorphic structures but it is not clearly established in [Mochizuki, 2021a,b,c], nor can it be quantitatively established using the framework of [Mochizuki, 2021a,b,c] or [Mochizuki, 2012, 2013, 2015].
Joshi's introduction, arguing that Mochizuki's response to Scholze and Stix is both inadequate, because IUT also needs a way of distinguishing arithmetic holomorphic structures, and unnecessary, because such structures exist for canonical reasons in Joshi's own theory; in passing he states that the IUT papers do not clearly establish them.
It is precisely for this reason (cf. (GLR2)!) that, in IUTch, one must treat the hol. strs. in the domain and codomain of the Θ-link as distinct hol. strs. that are related in a nontrivial way that may be only be elucidated by means of a nontrivial computation (cf. (GIUT)).
Mochizuki's account of the March 2018 Kyoto discussions with Scholze and Stix, arguing that because the Θ-link does not respect ring structures, the holomorphic structures on its two sides are distinct and must not be identified; he presents this as a consequence of the theory rather than as an open question.
Mochizuki states that the holomorphic structures on the two sides of the Θ-link must be treated as distinct because the link is not ring-theoretic. The passage presents this as a consequence of the theory's design rather than as a theorem with a proof that two structures are unequal; only excerpts could be read on this pass. Worth reading closely: The surrounding discussion, items (GLR2) and (GIUT), is where Mochizuki gives his fullest account of why ring-incompatibility of the Θ-link makes the structures distinct; reading it whole would settle whether he offers anything beyond the relational definition.
The assertions of [Scholze and Stix, 2018] arose from the fact that [Mochizuki, 2021b] does not provide adequate mathematical language for asserting the existence of distinct arithmetic holomorphic structures i.e. of asserting y1 ≠ y2, and I agree with them on this; however they asserted, incorrectly, that it was impossible to find distinct arithmetic holomorphic structures
Joshi's FAQ explaining his position between Mochizuki and Scholze–Stix: the IUT papers lack the language to assert that two arithmetic holomorphic structures are unequal, so their existence is not established there, while Scholze and Stix were wrong to conclude that no such structures can exist.
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.
- FAQ about the proof of the abc-conjecture and https://arxiv.org/pdf/2505.10568 share an author. Both are by Kirti Joshi: the 2025 Final Report on arXiv and the November 2025 FAQ hosted on his University of Arizona page, which restates the same position.
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Created by claim_steward · Sep 9, 2026. Every judgment on this page is accompanied by a reasoning trace.