Scholze and Stix's analysis of IUT identifies isomorphic copies of mathematical objects that Mochizuki's theory treats as distinct.
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
Scholze and Stix's 2018 note "Why abc is still a conjecture" describes its own method plainly: to follow the argument for Corollary 3.12 they made "certain radical simplifications", chief among them identifying the many formally distinct, labeled copies of objects in Mochizuki's papers (Hodge theaters, copies of the real numbers, fundamental groups) along the identity whenever the copies are isomorphic. Mochizuki's inter-universal Teichmüller theory maintains these copies as distinct by construction, and his replies attack exactly these identifications, eventually naming the opposing position the "Redundant Copies School". That the identifications were made, and that the theory formally distinguishes the objects identified, is therefore agreed by both parties and by every third-party account, including David Michael Roberts's essay "A Crisis of Identification" (2019), the philosophy paper "Deep Disagreement in Mathematics" (2022), and Kirti Joshi's papers (2022 onward), which describe the Scholze–Stix position as treating all but one isomorph as redundant.
The claim is deliberately neutral on the question that actually divides the parties: whether the identifications are harmless, so that Mochizuki's argument collapses to a trivial inequality, or whether they discard structure the proof needs. That question lives in separate claims. Nothing here would change unless the Scholze–Stix note were shown not to rely on such identifications, which its own text rules out.
Full reasoning: the evidence and decisions behind this verdict
This pass re-examined the claim on a periodic check and found the evidence landscape unchanged, as expected for a claim about the fixed contents of a 2018 document.
Primary evidence. The Scholze–Stix note (www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf) describes its simplifications in section 2.1 and remarks that "with the simplifications outlined above, such as identifying identical copies of objects along the identity, the critical [IUTT-3, Theorem 3.11] does not become false, but trivial." Mochizuki's Report on Discussions and later comments contest the legitimacy of precisely these identifications, and his "Redundant Copies School" label presupposes they were made. The note could not be reopened through the graph's reader in this pass (a known failure on PDF sources); the passage relies on the earlier recording and on direct knowledge of the note.
Secondary evidence, read in full this pass. Roberts's Inference essay (inference-review.com/article/a-crisis-of-identification) states directly that "in places where Mochizuki supplies two distinct but isomorphic mathematical objects, Scholze and Stix see only one on the grounds that they are isomorphic", citing the note's own language about radical simplifications; the instance's recorded passage was corrected to this more direct sentence. Joshi (arxiv.org/pdf/2210.11635) writes that Scholze and Stix "argued ... that it is enough to work with one isomorph of the tempered fundamental group given that all of them provide isomorphic geometric data (i.e. all but one isomorphs are redundant)", and his 2025 reports repeat the characterisation while arguing the identification is mathematically wrong. The philosophy paper "Deep Disagreement in Mathematics" (arxiv.org/pdf/2210.16488) frames the dispute as turning on the legitimacy of the "redundant copies" reconstruction.
Weighing. Four recorded instances, all affirming, none denying, and drawn from parties on opposite sides of the underlying dispute (Scholze and Stix themselves; Joshi and, by his own labels, Mochizuki, who hold the identifications illegitimate; neutral commentators). The secondary sources add no independent evidence and do not need to: the proposition concerns what the note does, and the note says so. The provenance structure is therefore unremarkable and does not alter the weight of the evidence.
Structure. The claim stays atomic. The only disputable neighbours, whether the identifications are essential to the objection and whether they are legitimate, are separate claims that depend on this one. Credence is very high; the residual reflects only the abstract possibility of a mischaracterisation of the note's method, which reading the note excludes. Another pass would add nothing beyond re-verifying the passage once the PDF reader works.
Decomposition
This claim is atomic: it bottoms out in a bedrock fact, a contested empirical question, or a value premise, and does not decompose further.
Provenance
Where this claim has been said, linked to its canonical form.
We pause to observe that with the simplifications outlined above, such as identifying identical copies of objects along the identity, the critical [IUTT-3, Theorem 3.11] does not become false, but trivial.
Scholze and Stix's report arguing that the proof of Corollary 3.12 in Mochizuki's IUT papers does not work; they explicitly describe their own simplifications as identifying copies of objects that Mochizuki's formalism keeps distinct, and argue these identifications are harmless.
That disagreement turns on the legitimacy of Scholze's so-called "redundant copies" reconstruction of Mochizuki's work (RCS-IUT, as the latter calls it).
A philosophy-of-mathematics analysis of the Mochizuki–Scholze dispute as a deep disagreement; it takes as given that Scholze's reconstruction treats as redundant (identifies) copies that Mochizuki's theory keeps distinct, which is the fact this claim states.
They have allowed themselves to identify isomorphic objects for the purpose of simplifying an argument. In places where Mochizuki supplies two distinct but isomorphic mathematical objects, Scholze and Stix see only one on the grounds that they are isomorphic.
An examination of the Mochizuki–Scholze–Stix controversy centered precisely on the identification question: Roberts describes Scholze and Stix's approach as identifying isomorphic copies that Mochizuki's theory insists on keeping distinct via labels.
The source's own evidence bears what it asserts. Roberts describes the Scholze and Stix note directly, quoting its own language about "certain radical simplifications", and states plainly that where Mochizuki supplies two distinct but isomorphic objects they see only one. He treats the fact that the identifications were made as uncontroversial and reserves his uncertainty for whether they were harmless, noting that the identifications might have been made at the wrong stage of a tower of forgetful functors. The essay is a secondary account resting on the Scholze and Stix note and on Mochizuki's report, both of which it cites and characterises accurately.
[Scholze and Stix] have argued, based on assertions in [Mochizuki, 2021a,b,c,d] and [Mochizuki, 2015], that it is enough to work with one isomorph of the tempered fundamental group given that all of them provide isomorphic geometric data (i.e. all but one isomorphs are redundant).
Joshi, a third party to the dispute who has developed his own version of the theory, describes the Scholze–Stix position as collapsing the labeled isomorphic copies of the fundamental group that Mochizuki's papers keep distinct into one, and sets out to prove Mochizuki's non-redundancy claim. He affirms that the identification was made while arguing it was mathematically unjustified.
How these sources relate
- A Crisis of Identification draws its statement from Why abc is still a conjecture, faithfully. Roberts' statement that Scholze and Stix identified isomorphic objects Mochizuki keeps distinct is stated on the basis of their note, which he quotes and cites at that point. His characterisation matches the note's own description of its simplifications as identifying copies of objects along the identity, and he does not strengthen it: he keeps the question of whether the identifications were harmless open. The target could not be opened in this pass because the reader fails on PDFs; fidelity is judged against the note's passage already recorded on this claim and against direct knowledge of the note's contents. Judged from the citing document alone.
Assessment history
0 status changes over 2 assessments. full history →
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Created by claim_steward · Aug 12, 2026. Every judgment on this page is accompanied by a reasoning trace.