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5. The critical transition: IUT III, Theorem 3.11 → Corollary 3.12

5.0 Reading map: the contested inference, both sides, what remains open

Read this section first. Everything from §5.1 onward is backing detail — exact quotes, page numbers, and each side's argument in full — for the claims made here. This file does not adjudicate who is right (no sentence below is a verdict), does not re-derive the ~150 pages of IUT III construction that precede Theorem 3.11 (term scope: 07-dictionary.md; claim graph: 00-map.md), and does not audit the Lean formalization effort in depth (see 06-lean-boundary.md and §5.9; that cross-reference sits in a shared, actively-edited directory — §5.12, flag 7).

The precise contested inference. Corollary 3.12 ("Log-volume Estimates for Θ-Pilot Objects," hosted-PDF pp.173–174 — pagination warning in §5.3) is derived from Theorem 3.11 ("the main theorem of the present series," p.153) through an eleven-step proof. The dispute concerns only the last step, Step (xi), where an abstract comparison licensed by Theorem 3.11 becomes a numerical log-volume inequality. Scholze–Stix ("SS," SS report p.9) and Mochizuki (Cmt2018-08 pp.3–4) agree that Step (xi) turns on identifying several a priori distinct copies of the real numbers (arising from objects relative to different "arithmetic holomorphic structures" linked by the Θ-link), mediated by indeterminacies the paper calls (Ind1), (Ind2), (Ind3), while tracking a scalar factor of \(j^2\).

Side-by-side (full citations: §5.4–§5.6):

Scholze–Stix Mochizuki
The Step (xi) map Must effectively be linear for the diagram to stay consistent; keeping the \(j^2\) scalar then forces an "empty inequality" Non-linear once indeterminacies are respected; the linearity SS assume ("(Lin)," Mochizuki's own label) is "completely false"
Effect on Theorem 3.11 Under SS's reading, Thm. 3.11 itself "does not become false, but trivial" Under SS's reading, Thm. 3.11's multiradial algorithms no longer apply at all — inapplicable, not trivialized
Overall verdict "There is no proof"; not fixable by small modifications Reflects "a fundamental misunderstanding" of what IUTch's objects are

What remains open (full list: §5.11): - whether Mochizuki's "(Lin)" label fairly restates what SS's own diagram assumes, or recharacterizes it; - whether SS's quoted Step-(xi) conclusion and IUT III's own boxed Corollary 3.12 are visibly the same inequality (§5.10, item 6) — not resolved here; - no public, dated SS reply specifically to Mochizuki's Sept. 2018 Cmt2018-08 was located; - what, beyond pagination, changed in IUT III's text between the version SS cite (Cor. 3.12 "at page 16") and the hosted 2020-05-18 PDF used throughout this file (p.173).

Evidence labels follow 00-map.md: STANDARD, ASSERTED_IN_IUT, DISPUTED, CONDITIONAL_FORMALIZATION, UNVERIFIED. Source-check labels follow sources.md: PDF_SECTION, SECONDARY; (CHECKED, 2026-10-05) marks a passage read directly in this research. PDF page is distinguished from printed page throughout (per 08-work-queue.md).

5.1 Sources consulted, and live access status (checked 2026-10-05)

# Document Self-declared date URL Access status (checked 2026-10-05)
1 Scholze & Stix, Why abc is still a conjecture Title page: "July 16, 2018" https://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdf Fetched and extracted in full earlier in this research pass (CHECKED). A later re-check hit a transient TLS/connection failure on this one server; since full text was already obtained and is internally consistent (page numbers, footnotes, references all resolve), this is flagged as a possible transient outage, not asserted as a permanent access loss
1a — same document, as hosted by Mochizuki, labelled "[SS2018-08] August 2018 Report" n/a https://www.kurims.kyoto-u.ac.jp/~motizuki/protectedpdf-2018-08/SS2018-08.pdf HTTP 403 (access-restricted), confirmed live
1b — the URL originally linked by Quanta Magazine in Sept. 2018 for the same document n/a http://www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-08.pdf (no protectedpdf- folder) HTTP 404 today. Confirmed: this exact URL is quoted live in Peter Woit's Sept. 2018 blog post (§5.9 below) as "[their write-up]... is now available here", so the file was freely reachable at this path in 2018 and has since been moved/restricted. The Bonn mirror (row 1) is the only copy this research could read
1c Earlier draft, "[SS2018-05] May 2018 Report" n/a https://www.kurims.kyoto-u.ac.jp/~motizuki/protectedpdf-2018-05/SS2018-05.pdf HTTP 403. Not independently obtained; its content is known here only through Mochizuki's quotations of it in Cmt2018-05
2 Mochizuki, Report on Discussions, held during the period March 15–20, 2018, concerning Inter-universal Teichmüller Theory (IUTch) ("Rpt2018"), with the cooperation of Yuichiro Hoshi Title page: "February 2019"; hub page says "updated on 2019-02-01" https://www.kurims.kyoto-u.ac.jp/~motizuki/Rpt2018.pdf HTTP 200, fetched and extracted in full (45 pp.) (CHECKED)
3 Mochizuki, Comments on the Manuscript by Scholze–Stix... ("Cmt2018-05") Title page: "July 2018" (PDF container metadata separately shows Sept. 2018 — a minor, unresolved discrepancy, not pursued further here) https://www.kurims.kyoto-u.ac.jp/~motizuki/Cmt2018-05.pdf HTTP 200, fetched and extracted in full (CHECKED)
4 Mochizuki, Comments on the Manuscript (2018-08 version) by Scholze–Stix... ("Cmt2018-08") Title page: "September 2018"; opens by saying it supplements Cmt2018-05 and responds to "the August 2018 version of the manuscript [SS2018-08]" https://www.kurims.kyoto-u.ac.jp/~motizuki/Cmt2018-08.pdf HTTP 200, fetched and extracted in full (CHECKED)
5 Mochizuki's hub page for the whole exchange n/a https://www.kurims.kyoto-u.ac.jp/~motizuki/IUTch-discussions-2018-03.html HTTP 200 (redirects http→https), fetched in full (CHECKED)
6 Mochizuki, Inter-universal Teichmüller Theory III: Canonical Splittings of the Log-theta-lattice — currently hosted preprint PDF Publications list marks this file "NEW!! (2020-05-18)" https://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20III.pdf HTTP 200, fetched and extracted in full (199 pp. in this extraction) (CHECKED)
7 IUT I–IV, journal version Special issue dated 2021-03-05 PRIMS 57 (1/2), DOIs 10.4171/PRIMS/57-1-1…-4, pp. 3–207, 209–401, 403–626, 627–723 Issue page and bibliographic listing resolved; DOI -3 individually resolved with matching abstract in earlier research. -1,-2,-4 confirmed via the publisher's own issue listing, not each individually re-fetched article-by-article in this pass
8 Klarreich, "Titans of Mathematics Clash Over Epic Proof of ABC Conjecture," Quanta Magazine 2018-09-20 (date in URL slug, confirmed) https://www.quantamagazine.org/titans-of-mathematics-clash-over-epic-proof-of-abc-conjecture-20180920/ HTTP 200 (CHECKED)
9 Woit, "Scholze and Stix on the Mochizuki Proof," Not Even Wrong Undated in the fetched text; contemporaneous with, and links to, the Quanta piece (so ≈ Sept. 2018) https://www.math.columbia.edu/~woit/wordpress/?p=10560 HTTP 200 (CHECKED)
10 Woit, "Million Dollar Prize for Scholze and Stix," Not Even Wrong ≈ July 2023 (references a Tokyo news conference "today" and links a Nikkei piece from that period) https://www.math.columbia.edu/~woit/wordpress/?p=13573 HTTP 200 (CHECKED)
11 Castelvecchi, "Maths proof that rocked number theory will be published," Nature 580, p.177 2020-04-09; reports a press conference 2020-04-03 https://www.nature.com/articles/d41586-020-00998-2, DOI 10.1038/d41586-020-00998-2 The canonical URL returns HTTP 406 to this session's text-fetching tool specifically (a header/content-negotiation issue, since a plain HEAD request to the same URL succeeds with HTTP 200); full text obtained instead from a Nature-hosted PDF mirror and extracted (CHECKED)
12 New Scientist, "Decade-long struggle over maths proof could be decided by $1m prize" ≈ July 2023 https://www.newscientist.com/article/2381521-decade-long-struggle-over-maths-proof-could-be-decided-by-1m-prize/ HTTP 200 (CHECKED)
13 New Scientist, article 2424640 (Joshi/Scholze/Mochizuki/Fesenko) 2024-03-28 newscientist.com/article/2424640 HTTP 200 (CHECKED earlier in this research)
14 New Scientist, article 2522687 (LANA "stuck point," Topaz, Buzzard) 2026-04-10 (print issue 2026-04-18) newscientist.com/article/2522687 HTTP 200 (CHECKED earlier in this research)
15 ZEN University / IUGC press releases on the IUT Innovator/Challenger Prizes 2023-07-07 (prize creation); first award reported ≈ April 2024 zen.ac.jp/news/0ul6zqed9-0; zen.ac.jp/news/d-5ye560_l HTTP 200 (CHECKED)
16 LANA (lana-agents/iut) GitHub README n/a raw.githubusercontent.com/lana-agents/iut/... HTTP 200. Full depth already audited in 06-lean-boundary.md; used here only for the one-paragraph cross-reference in §5.9

No claim below relies on a source this research could not actually open. Where a source could not be opened (rows 1a, 1b, 1c), that fact is reported, not papered over.

5.2 Established-fact timeline

These are dated, source-confirmed events, not characterizations of who is right. Dated public commentary (who said what) is collected in §5.9, not repeated here.

Date Event Source
2012 Mochizuki posts four IUT preprints online Nature 2020 (row 11); Mochizuki's publications page
2018-03-15 to 2018-03-19 Scholze and Stix spend a week at RIMS, Kyoto, discussing the proof with Mochizuki and Hoshi Rpt2018 title page (p.1); SS report p.1
2018-05 SS circulate a first manuscript ("[SS2018-05]") Hub page (row 5); Cmt2018-05
2018-07-16 (self-dated) / called "August 2018" by Mochizuki's site SS circulate/finalize the manuscript still hosted today ("[SS2018-08]"), Why abc is still a conjecture SS report title page; Cmt2018-08 p.1 — the July/August discrepancy is unresolved; see §5.12
2018-07 and 2018-09 Mochizuki posts Cmt2018-05 and Cmt2018-08, responding point-by-point Title pages of each document
2018-09 Quanta (Klarreich) publishes an account with direct quotes from both sides; Woit's blog independently summarizes the dispute Rows 8–9
2019-02 (self-dated) Mochizuki finalizes/updates Rpt2018 Rpt2018 title page; hub page
2020-04-03 Kyoto press conference announces PRIMS's acceptance of IUT I–IV, given by Kashiwara and Tamagawa (not Mochizuki) Nature 2020, p.1
2020-04-09 Nature publishes Castelvecchi's report, including that Mochizuki is PRIMS's chief editor Row 11
2020-05-18 The IUT III preprint PDF used throughout this file is last marked "NEW!!" on Mochizuki's publications list — after the 2018 SS exchange, hence the pagination gap (§5.12) Mochizuki's publications-list page (row 6)
2021-03-05 IUT I–IV appear in print, PRIMS vol. 57 no. 1/2 DOIs (row 7)
2022–2024 A Mochizuki-coauthored extension paper appears (Kodai Math. J. 45); IUGC (ZEN University, est. 2023-06-06) announces IUT-related prizes on 2023-07-07, including a USD 1,000,000 award for a published disproof; a separate, smaller prize is first awarded ≈2024-04 for the 2022 paper — detail in §5.9 Rows 10, 12, 15
2023–2026 Two separate Lean-formalization efforts proceed (Mochizuki's own; LANA, Kato/Topaz); LANA reports in 2026-04 reaching a point its lead calls "closely related" to SS's objection Row 14; 06-lean-boundary.md

5.3 What Theorem 3.11 and Corollary 3.12 assert, and the exact relation between them ASSERTED_IN_IUT

⚠ Pagination warning, read before using any page number below: the Scholze–Stix report cites the corollary's statement at "[IUTT-3, page 16, Corollary 3.12]" (SS report, printed p.4). In the IUT III PDF as currently hosted (row 6, a file last updated 2020-05-18, i.e. roughly two years after SS wrote), Theorem 3.11 begins at printed/PDF page 153 and Corollary 3.12 at printed/PDF page 173. This is not an error in either document; it is direct, dated evidence that the hosted text of IUT III has been substantially revised (almost certainly lengthened, e.g. with new discussion and the "invisible indeterminacies" Figures 3.4–3.5 that now immediately precede Corollary 3.12) between whatever version SS read in 2018 and the version read for this file in 2026. All page numbers below are to the 2020-05-18 hosted PDF and are not stable identifiers across versions; the theorem/corollary numbers are the stable identifiers.

Theorem 3.11 ("Multiradial Algorithms via LGP-Monoids/Frobenioids," IUT III p.153) is introduced by the paper's own text, one sentence before its statement, as "the main theorem of the present series of papers" (p.153). Fixing a collection of Hodge theaters arising from a log-theta-lattice, part (i) is labelled "Multiradial Representation." Per the paper's own summary in Remark 3.11.1(i) (pp.159–160), the theorem gives an algorithm for describing the LGP-monoids of one node of the log-theta-lattice in terms of the "a priori alien arithmetic holomorphic structure" of a different, Θ-link-linked node — up to three explicitly named indeterminacies, (Ind1), (Ind2), (Ind3).

Corollary 3.12 ("Log-volume Estimates for Θ-Pilot Objects," IUT III pp.173–174) is introduced, one sentence before its statement, as

"a relatively concrete consequence of the somewhat abstract content of Theorem 3.11" (p.173).

Its statement opens "Suppose that we are in the situation of Theorem 3.11" (p.173) — i.e. it does not stand on its own; it shares Theorem 3.11's hypotheses and directly consumes Theorem 3.11's "multiradial representation" as an ingredient. Its headline conclusion, quoted exactly from the PDF (p.174):

"\(C_\Theta \ge -1\) for any real number \(C_\Theta \in \mathbb{R}\) such that \(-|\log(\Theta)| \le C_\Theta \cdot |\log(q)|\)."

The exact relation between the two (per IUT III's own proof structure, not a paraphrase): Corollary 3.12's proof is a named multi-step argument (Steps (i) through (xi), per the paper's own labels, spanning roughly pp.174–195 of the hosted PDF). The quantities "\(-|\log(\Theta)|\)" and "\(-|\log(q)|\)" are procession-normalized mono-analytic log-volumes of specific regions ("holomorphic hulls" of pilot-object images) defined using Theorem 3.11's apparatus (p.173). The multiradial algorithm from Theorem 3.11 is explicitly invoked again inside this proof — SS's own report cites this as "the multiradial algorithm [IUTT-3, Theorem 3.11]" being applied "so... it is argued in [IUTT-3, Corollary 3.12]" (SS report p.9) — and the disputed derivation step is specifically Step (xi), the last step, where the abstract multiradial comparison is converted into the numerical log-volume inequality above. Both SS and Mochizuki agree on this locus: SS's report says the issue arises "towards the end of Step (xi) in the proof of [IUTT-3, Corollary 3.12]" (SS report p.9), and Mochizuki's replies repeatedly cite the same Step (xi) (Cmt2018-08 (C12)–(C14), pp.3–4).

A more nuanced point, worth stating precisely rather than compressing into "Cor 3.12 follows from Thm 3.11": SS's own footnote to their critique (SS report, p.9, footnote 12) states that under the identification they say consistency forces,

"the critical [IUTT-3, Theorem 3.11] does not become false, but trivial."

That is, SS's objection is not confined to "the step from 3.11 to 3.12 has a gap"; part of their claim is that, under a specific reading of how objects must be identified, Theorem 3.11 itself would carry no content. Mochizuki's response is not that this reading is a valid simplification that happens to trivialize a true theorem; it is that the reading is illegitimate for 3.11 in the first place — once SS's simplification is made, "one can no longer apply the multiradial algorithms of [IUTchIII], Theorem 3.11" at all (Cmt2018-08 p.4, comment (C13)). So the disagreement is one level more structural than a simple missing-step claim: the two sides disagree about what operation (identifying copies of the real numbers that arise relative to different arithmetic holomorphic structures) is legitimate to perform at all in Step (xi), and therefore about whether SS's simplified setting is a faithful reduction of the real argument or a different, inapplicable one. §5.6–5.7 develop this precisely.

5.4 The Scholze–Stix objection, in their own terms DISPUTED

Source: SS report (row 1), §2.2, "Proof of [IUTT-3, Corollary 3.12]," pp.9–10, unless noted.

  1. SS set up two Hodge theaters \(HT_1, HT_2\) linked by a Θ-link, under which "the abstract Θ-pilot object from \(HT_1\) is mapped to the abstract \(q\)-pilot object belonging to \(HT_2\)" (p.9).
  2. They identify that the derivation of the headline inequality requires comparing several distinct copies of 1-dimensional ordered \(\mathbb{R}\)-vector spaces that arise in the construction — they list three families: (1) spaces where "abstract" pilot elements live, (2) spaces where "concrete" pilot elements live (one per index \(j=1,\dots,\ell^\star\) on the Θ-side, one on the \(q\)-side), (3) two further copies of the standard reals \(\mathbb{R}^\Theta, \mathbb{R}^q\) where "arithmetic degrees" live (p.9). They draw this out as an explicit diagram (p.9–10) of maps between these copies.
  3. They state that reaching the headline inequality additionally requires rescaling by a factor of \(j^2\) for each index \(j\) (p.4, their equation (1.5); p.9, "it was necessary to change the isomorphism \(\mathbb{R}\cong\mathbb{R}^{\odot,\Theta}\) by the scalar \(j^2\)"), in order for the abstract Θ-pilot object to "encode the arithmetic degree of the (\(j\)-th) concrete Θ-pilot object" (p.9).
  4. Their central claim: if one insists on consistently identifying all the copies of \(\mathbb{R}\) in the diagram using the natural isomorphisms they describe, introducing the \(j^2\) scalars "strictly speaking leads to inconsistencies, i.e. monodromy" (p.10) — and the only way to keep the diagram consistent is to omit the \(j^2\) scalars, "which leads to an empty inequality" (p.10, their emphasis).
  5. They report that, in the March 2018 discussions, Mochizuki responded that the diagram commutes only "up to the 'blurring' given by certain indeterminacies"; SS's own gloss on this response is: "it seems to us that this statement means that the blurring must be by a factor of at least \(O(\ell^2)\) rendering the inequality thus obtained useless" (p.10).
  6. As noted in §5.3, their footnote 12 (p.9) separately records that, under the "identifying identical copies of objects along the identity" simplification, Theorem 3.11 itself "does not become false, but trivial."
  7. SS's own overall verdict, stated on p.1, before any of the technical argument: "We, the authors of this note, came to the conclusion that there is no proof," calling the problem "so severe that in our opinion small modifications will not rescue the proof strategy" (p.1). They explicitly flag that they "supplement our report by mentioning dissenting views from Prof. Mochizuki and Prof. Hoshi" (p.1).

All of the above is SS's own claim, in SS's own terms, as published by SS. It is reported here as a claim, not re-derived or checked line-by-line against the full IUT III apparatus in this file (see §5.10 for what that independent check would require).

5.5 Mochizuki's counter-interpretation, in his own terms DISPUTED

Source: primarily Cmt2018-08 (row 4), comments (C12)–(C14), pp.3–4; Rpt2018 (row 2), §§10–12 and the compact summary (Smm), p.2.

  1. Mochizuki frames the dispute as turning on "the issue of distinguishing the abstract category-theoretic versions of pilot objects... from their concrete (multiradial!) representations," which he calls "one of the most central aspects of IUTch," citing Theorem 3.11 and the proof of Corollary 3.12 by name (Cmt2018-08, (C12), p.3).
  2. He states that SS's simplifications (identifying objects "along the identity") correspond to what he calls the "id-version" of the construction, discussed at length in Rpt2018 §10, and that once this simplification is made, "one can no longer apply the multiradial algorithms of [IUTchIII], Theorem 3.11" (Cmt2018-08, (C13), p.4) — i.e. in his account the simplified setting SS analyze is not a faithful, trivialized restriction of the real argument, but a setting in which the real argument's tools no longer apply at all.
  3. He isolates what he presents as SS's governing assumption and labels it (Lin) — his own term, not SS's: any two such copies of \(\mathbb{R}\) are related simply by multiplication by some scalar (Cmt2018-08, (C14), p.4).
  4. His response: assumption (Lin) "is completely false" — where indeterminacies are involved, the relationship between log-volumes is geometry-dependent and "highly non-linear" (Cmt2018-08, (C14), p.4), illustrated by the worked example (LVEx) in §5.8.
  5. His short, compact version of this whole disagreement (Rpt2018, (Smm), p.2) models the structure of the argument as: positive reals \(A,B\) with \(-2B=-A\) (standing for the Θ-link), a theorem \(-2B\le -2A+1\) (standing for the multiradial representation of Theorem 3.11), jointly implying \(A\le 1\) (standing for Corollary 3.12). He states that SS's misunderstanding, on this model, amounts to assuming "the theory remains essentially unaffected even if one takes \(A=B\)" — which forces \(A=B=0\), a contradiction — and states explicitly that "the essential content... of IUTch fail(s) to hold under the assumption 'A=B'," so the contradiction "does not imply the existence of any flaws whatsoever in IUTch" (Rpt2018, p.2).
  6. He asserts, as a matter of process, that "IUTch has been checked, verified, read and reread, and orally exposed in detail in seminars in its entirety countless times" since 2012 by a group of mathematicians, citing Fesenko's estimate ("verified at least 30 times") (Rpt2018, (Vrf1), pp.42–43), and that there is "no substantive mathematical reason whatsoever to suspect the existence of any oversights" (Rpt2018, (Vrf2), p.43).
  7. His own diagnosis: the dispute may arise because the mathematics SS understand by "IUTch" differs substantially from the mathematics he and colleagues understand by that name (Rpt2018, (DfMth), p.44).

All of the above is Mochizuki's own claim, in his own terms, as published by him. Like §5.4, it is reported, not independently re-derived here.

5.6 Side-by-side table

Point at issue Scholze–Stix (SS report, §2.2 unless noted) Mochizuki (Cmt2018-08 / Rpt2018 unless noted)
Where is the disputed step located? "Towards the end of Step (xi) in the proof of [IUTT-3, Corollary 3.12]" (p.9) Same locus, repeatedly cited: Step (xi) (Cmt2018-08 (C12)–(C14), pp.3–4)
What operation is at stake? Consistently identifying several copies of \(\mathbb{R}\) (abstract pilot, concrete pilot ×\(\ell^\star\), arithmetic-degree copies) across the Θ-link (p.9) Same operation, but described as subject to indeterminacies (Ind1)–(Ind3) belonging to the multiradial representation of Thm. 3.11 (Cmt2018-08 (C14), p.4)
What is the relationship between these copies, in each side's account? A "simple, straightforward linear relationship... multiplication by some positive real number" is implicitly required for a meaningful inequality (this phrase, "(Lin)," is Mochizuki's label for SS's assumption, not SS's own words) Labels the above assumption "(Lin)" and states it is "completely false"; the real relationship (once indeterminacies are accounted for) is "highly non-linear" (Cmt2018-08 (C14), p.4)
What happens to the disputed \(j^2\) scalar? Consistent identification forces it to be dropped, "which leads to an empty inequality" (p.10) Does not address the \(j^2\) computation on SS's own terms; reframes the whole computation as illegitimate once SS's "id-version" identifications are made (Cmt2018-08 (C13), p.4)
What did Mochizuki say in the March 2018 discussions themselves, per SS? He said the diagram commutes "up to the 'blurring' given by certain indeterminacies"; SS gloss this as implying a useless bound, "at least \(O(\ell^2)\)" (p.10) Not separately re-stated in these terms in the later written comments; the later comments instead refer to Rpt2018's own worked analogy (LVEx) as the general explanation
Does the objection bear only on Cor. 3.12, or on Thm. 3.11 too? Footnote 12 (p.9): under their reading, "the critical [IUTT-3, Theorem 3.11] does not become false, but trivial" (C13), p.4: under SS's simplification, "one can no longer apply the multiradial algorithms of [IUTchIII], Theorem 3.11" at all — i.e., not a trivialization of the real theorem, but inapplicability of it
What does each side conclude about the proof overall? "We, the authors of this note, came to the conclusion that there is no proof" (p.1); "a problem so severe that... small modifications will not rescue the proof strategy" (p.1) The objection reflects "a fundamental misunderstanding" (Cmt2018-08 (C11)–(C12), p.3) stemming from SS using a different, non-equivalent mathematical setting than the one IUTch actually defines (Rpt2018 (DfMth), p.44)
What does each side say about the state of outside verification? Not a topic of the SS report itself, which is a mathematical note, not a survey of community opinion Cites the testimony of "colleagues involved" and Fesenko's "verified at least 30 times" estimate as evidence against the existence of a flaw (Rpt2018 (Vrf1)–(Vrf2), pp.42–43)

5.7 Locus of unresolved disagreement (neutral synthesis)

This section states, as precisely as the two primary documents allow, where the live disagreement sits — without deciding it.

Both sides agree on: - the location (Step (xi) of the proof of Corollary 3.12); - that the step requires comparing several a priori distinct copies of the real numbers, arising from objects relative to different "arithmetic holomorphic structures" linked by the Θ-link; - that the comparison is mediated by indeterminacies the paper calls (Ind1), (Ind2), (Ind3) and that these indeterminacies are not optional — SS's own report acknowledges they are present ("up to certain indeterminacies, e.g. (Ind1,2,3) (without which the conclusion would be obviously false)", SS report p.9).

The documents diverge on: - whether the specific comparison performed in Step (xi) is linear or non-linear in the relevant sense, and consequently whether the \(j^2\) scalar can be consistently retained; - what follows if one insists on the "consistent identification" reading SS use: SS read it as revealing an empty/useless inequality in Cor. 3.12 (and triviality in Thm. 3.11 under the same reading); Mochizuki reads the insistence on that reading itself as the error, because, in his account, it is not a special/simplified case of the multiradial representation but a different construction to which Theorem 3.11 does not apply; - whether the disagreement is a checkable local computation or a disagreement about which mathematical object the words denote: Mochizuki's own diagnosis (Rpt2018 (DfMth), p.44) explicitly frames it as the latter — that SS's "IUTch" and his "IUTch" denote different mathematical content — which, if accurate, would mean the dispute cannot be settled merely by checking arithmetic inside a single shared formalism, because the two sides would not (on this account) agree on what is being formalized. SS's report does not address this particular framing; it presents its criticism purely as an internal computation within IUT III's own stated objects and definitions.

Neither primary document, read on its own, resolves this. Each gives a self-consistent account of why the other side is mistaken. Readers wanting to go further than this file should attempt the checklist in §5.10, and should note the asymmetry in available independent (non-participant) secondary commentary recorded in §5.9 and flagged again in §5.12.

5.8 A worked schematic analogy

⚠ This is explicitly NOT part of IUT. It is Mochizuki's own illustrative toy model, offered by him as an elementary analogy for one qualitative phenomenon (indeterminacies producing a non-linear relation between log-volumes), not as a proof, derivation, or substitute for any part of IUT III. Mochizuki himself appends an explicit limiting caveat, quoted at the end of this section, that the analogy's internal consistency is special to the specific numbers/regions chosen and does not generalize to arbitrary regions and constants. Source: Rpt2018 (row 2), item (LVEx), pp.24–26 (preceded by (LbLV), p.23, and (MlLV), p.24). All arithmetic below was independently recomputed for this file by direct area calculation and reproduced to match the source exactly.

The construction, exactly as given. Let \(V=\mathbb{R}^2\) and let \(\sigma: V\to V\) be \(\sigma(x,y)=(-x,y)\), an order-2 automorphism. Let \(W\) be the stack-theoretic quotient of \(V\) by the group \(G=\{1,\sigma\}\), giving a degree-2 finite étale map of orbispaces \(\varphi: V\to W\). For positive reals \(a,b\), define two regions of \(V\):

\[ R_{a,b} = \{(x,y): |x|\le 1,\ 0\le |y|\le a\}\ \cup\ \{(x,y): 0\le x\le 1,\ a\le|y|\le a+b\}, $$ $$ S_{a,b} = R_{a,b}\cup\sigma(R_{a,b}). \]

\(R_{a,b}\) is an "inner band" of full width \(2\) and height \(a\), plus an "outer band" of half-width \(1\) and height \(b\); \(S_{a,b}\) reflects (doubles) only the asymmetric outer band. Writing \(\mu_{\log}\) for the natural log of ordinary Euclidean area:

\[ \mu_{\log}(S_{a,b}) = \log(4a+4b) \;>\; \mu_{\log}(R_{a,b}) = \log(4a+2b). \]

(Independently checked: area of \(R_{a,b}\) is \(2\cdot(2a) + 1\cdot(2b) = 4a+2b\); \(S_{a,b}\) doubles only the outer band to full width, giving \(4a+2(2b)=4a+4b\). The source's own stated inequality is confirmed exactly.)

Since \(\sigma(S_{a,b})=S_{a,b}\), \(S_{a,b}\) is "defined over \(W\)" (label \(W\)); since \(\sigma(R_{a,b})\ne R_{a,b}\), \(R_{a,b}\) is only "defined over \(V\)" (label \(V\)) — i.e. \(R_{a,b}\) is pictured as an "incomplete portion" of the \(G\)-symmetric object whose log-volume is read off from \(S_{a,b}\).

Now pick \(\lambda\in\mathbb{R}\) with \(\mu_{\log}(S_{a,b}) > 0 > \lambda > \mu_{\log}(R_{a,b})\) (the source notes concrete values of \(a,b,\lambda\) exist satisfying this — independently confirmed here, e.g. \(a=0.1,b=0.2\) gives \(\mu_{\log}(R_{a,b})=\log(0.8)\approx-0.223\), so any \(\lambda\in(-0.223,0)\) works, with \(\mu_{\log}(S_{a,b})=\log(1.6)\approx0.47>0\)). This makes \(\rho:=\mu_{\log}(R_{a,b})/\lambda > 1\) well-defined, and \(\lambda\) is read as the log-volume of a region \(R_\lambda\) "defined over \(W\)."

The correspondence table, exactly as the source states it (Rpt2018 p.25):

Toy-model object Corresponds to (per Mochizuki)
\(R_{a,b}\) the Θ-pilot object, relative to the Θ-holomorphic structure
\(S_{a,b}\) the holomorphic hull, relative to the \(q\)-holomorphic structure, of the multiradial representation of the Θ-pilot, with "indeterminacies" given by the action of \(G\)
\(R_\lambda\) the \(q\)-pilot object
the assignment \(\mu_{\log}(R_{a,b})_V \mapsto \lambda_W\) an \(\mathbb{R}\)-linear gluing isomorphism \(\mathbb{R}_V\xrightarrow{\sim}\mathbb{R}_W\) (dividing by \(\rho\)), corresponding to the Θ-link
the chain \(\mu_{\log}(S_{a,b})_W > \mu_{\log}(R_{a,b})_V \mapsto \lambda_W\) the chain of log-volume relationships in the argument of Step (xi) of the proof of Corollary 3.12

The point Mochizuki draws from it: \(\mu_{\log}(S_{a,b})_W > \mu_{\log}(R_{a,b})_V\) follows from the geometry of \(R_{a,b}\subseteq S_{a,b}\) and the stack-quotient construction of \(W\) — not from the linear gluing map — yet stays "entirely logically consistent" with that gluing (Rpt2018, pp.25–26). The model is built so a non-linear, geometry-dependent fact and a linear isomorphism coexist without contradiction, illustrating how assuming only linearity (as in (Lin)) could wrongly predict a contradiction or a vacuous inequality.

Mochizuki's own explicit limit on the analogy: he states directly that this consistency property does not hold "by arbitrary regions of \(W\), \(V\) and an arbitrary real number \(\lambda\)" (Rpt2018, p.26) — the example's internal consistency depends on the specific geometry and numeric inequalities chosen; it is one existence example of a qualitative phenomenon (non-linearity compatible with logical consistency), not a general theorem, and not a model of IUT III's actual objects beyond that one point. His own closing line calls Step (xi) "precisely the sort of situation" illustrated here (Rpt2018, p.26) — an analogy claim, not an identity claim.

A second, much more compact version of the same rhetorical move appears earlier in the same report as (Smm) (Rpt2018, p.2, quoted in §5.5, point 5, above): positive reals \(A, B\) with \(-2B=-A\), a theorem \(-2B\le-2A+1\), giving \(A\le1\); Mochizuki states SS's implicit error, on this model, is equivalent to assuming the theory is unaffected by setting \(A=B\) (which is false on his account and forces a contradiction \(A=B=0\) that he says reflects the invalidity of that assumption, not a flaw). This is an even smaller, purely-algebraic schematic and is likewise explicitly marked by Mochizuki as a "very rough" summary (Rpt2018, p.2), not a derivation.

5.9 Subsequent status: publication versus community acceptance

Established, dated facts (not characterizations):

  • IUT I–IV were accepted for publication in PRIMS, announced at a Kyoto press conference on 2020-04-03 by Masaki Kashiwara and Akio Tamagawa (Mochizuki did not attend) (Nature 2020, p.1).
  • Mochizuki is PRIMS's chief editor; PRIMS is published by RIMS, Kyoto University, where he works (Nature 2020, p.1). A secondary source (New Scientist, row 12) paraphrases Nature as reporting he had no personal role in the editorial decision on his own paper; this file could not independently confirm that specific sentence in the Nature text it obtained, so it is reported as New Scientist's characterization, not as independently verified primary text.
  • IUT I–IV were published in print in PRIMS 2021-03-05 (special issue 57(1/2)) — roughly eleven months after the acceptance announcement. Some secondary sources (e.g. New Scientist, row 12) describe the paper as "published in 2020," conflating the acceptance announcement with the later print date; this file keeps the two dates distinct.
  • A 2022 Kodai Math. J. paper by Mochizuki and four coauthors applies IUT to obtain explicit abc-type inequalities and a conditional new proof of Fermat's Last Theorem; it is an application/extension, not an independent proof addressing the Scholze–Stix objection. It was later recognized by an IUGC prize (≈2024) — IUGC being an institution dedicated to promoting IUT theory, not an independent mathematical society (ZEN press release, row 15).
  • A separately announced USD 1,000,000 "IUT Challenger Prize" requires a peer-reviewed paper meeting stated bibliographic eligibility rules (Woit, row 10) that an unpublished note such as the Scholze–Stix report does not on its face meet, independent of its mathematical content.
  • Two distinct Lean formalization efforts exist: LANA (Kato/Topaz, ZEN Mathematics Center, started late 2023, announced 2026-03-31) and a separate effort run by Mochizuki himself, which per New Scientist (row 14) is explicitly not aimed at independent verification. Full technical detail on either Lean effort is out of scope for this file — see 06-lean-boundary.md.

Characterizations by named individuals (dated, attributed, not adopted as fact by this file): immediately after the 2018 exchange, Stix described the proof as having a "serious, unfixable gap" while Scholze maintained abc "is still open" (both via Quanta, row 8). Woit, a self-described non-expert commentator, wrote in 2018 that Mochizuki's responses did not seem to "effectively address" the specific SS objection (row 9), and by 2023 characterized it as "generally accepted by experts" that the SS paper "conclusively shows" a flaw — a stronger, unhedged version of his earlier post (row 10). At the April 2020 acceptance announcement, Kedlaya said community opinion had not much changed and Tamagawa said review found "no fundamental alteration" needed (both row 11). In March 2024, Scholze reiterated the work "falls far short of giving a proof of ABC," while Mochizuki called Kirti Joshi's unrelated alternative work "mathematically meaningless" (row 13). In April 2026, LANA's lead Adam Topaz reported its formalization stuck at a point "closely related" to the SS objection (row 14). Separately, Fesenko's own survey (cited in Rpt2018, p.42, not independently checked by this file) estimates IUTch has been "verified at least 30 times" (row 2).

This research located substantially more independent (non-IUT-affiliated) commentary characterizing the objection as unresolved or credible than comparably independent commentary asserting it has been rebutted. This may reflect the actual state of published commentary, or gaps in this research; it is flagged, not treated as a finding about mathematical correctness. Most "verification" claims on the pro-IUTch side available to this research (Fesenko's count, the IUGC prize panel) come from parties actively involved in promoting or extending IUT — a fact about provenance, not a basis for discounting their content.

5.10 Reproduction checklist for a mathematical reader

To form an independent view (which this file deliberately does not do), a reader would need to, at minimum:

  1. Read IUT III's definitions of Θ-pilot object and \(q\)-pilot object (Definition 3.8, cited by both sides) and write down, in one's own notation, exactly which category each lives in and what data specifies an isomorphism between two instances.
  2. Read the full statement of Theorem 3.11, parts (i)–(iii) (IUT III pp.153–159 in the hosted version), and write down the precise scope of each of (Ind1), (Ind2), (Ind3) — what group, or what range of choices, each permits — not just their names.
  3. Read Corollary 3.12's full proof, Steps (i)–(xi) (pp.174–195 in the hosted version), and independently identify every point at which a "copy of \(\mathbb{R}\)" (in SS's language) or an "arithmetic holomorphic structure" (in Mochizuki's) is introduced, and what map/isomorphism is asserted to relate it to the others.
  4. At Step (xi) specifically, write out explicitly whether the map used to pass between the Θ-side and \(q\)-side log-volume is asserted to be: (a) literally linear everywhere it is used, (b) linear only outside the region affected by (Ind1)–(Ind3) and non-linear/indeterminate within it, or (c) some other relationship — and locate the exact sentence(s) of IUT III that settle this, rather than inferring it from either side's commentary.
  5. Independently verify the arithmetic of SS's equation (1.5) (SS report p.4) — the weighted sum over \(j=1,\dots,\ell^\star\) with coefficients involving \(j\) and \(j^2\) — and check whether dropping the \(j^2\) term, as SS say consistency requires, actually yields the "empty inequality" they describe, by the reader's own calculation rather than by trusting either side's characterization of the result.
  6. Independently reconcile SS's quoted intermediate conclusion from Step (xi), "\(-|\log(q)|\le -|\log(\Theta)|\in\mathbb{R}\)" (SS report p.9), with the final boxed statement of Corollary 3.12 as printed, "\(C_\Theta\ge-1\) for any \(C_\Theta\in\mathbb{R}\) such that \(-|\log(\Theta)|\le C_\Theta\cdot|\log(q)|\)" (IUT III p.174) — these are not obviously the same inequality on their face (one is a single fixed relation, the other parametrized by a free constant \(C_\Theta\)), and this file did not itself re-derive how one yields the other; doing so is a concrete, well-posed task for a reader, not resolved here.
  7. Obtain, if possible, the actual 2018 version of IUT III that SS cite as having Corollary 3.12 "at page 16" (not available to this research — see §5.12), and check whether the mathematical content of Theorem 3.11/Corollary 3.12 changed between that version and the 2020-05-18 version read for this file, or whether only surrounding discussion/numbering changed.
  8. Only after 1–7, read §5.4 and §5.5 above again and check which, if either, side's characterization matches what was found.

5.11 Open questions / unanswered concrete verification obligations

These are stated as questions, not rhetorical ones with an implied answer:

  1. Is the map used at the disputed point of Step (xi) linear, non-linear, or does the dispute itself partly consist in the two sides describing different maps as "the" map at that point? (§5.10.4 is the precise version of this question.)
  2. Does SS's "(Lin)" characterization (Mochizuki's label for their alleged assumption, not their own phrase) accurately represent what SS's own diagram and equations (SS report pp.9–10) actually assume, or does it simplify/recharacterize their argument in translation? Neither document read for this file settles this from the other side's perspective in a way phrased in common, shared notation.
  3. Is the "id-version"/simplified setting that Mochizuki says is the real content of SS's argument (Cmt2018-08 (C13), p.4) in fact what SS's report performs, or is this itself a disputed characterization of SS's argument? This file found no passage in which SS respond, in writing, specifically to the "id-version" label or to (Lin) by name (both are Mochizuki's post hoc labels for SS's argument, introduced in Mochizuki's replies) — i.e., no public, dated SS rebuttal of Mochizuki's Sept. 2018 Cmt2018-08 was located by this research. If one exists, it would bear directly on this file's open questions and was not found.
  4. What, precisely, changed in IUT III's text (not just pagination) between whatever version SS examined (page 16 for Cor. 3.12) and the 2020-05-18 hosted version (page 173)? Not established here (§5.12).
  5. Does either side's account change if the "indeterminacies" (Ind1)-(Ind3) are given a fully explicit, independent (non-IUT) group-theoretic or measure-theoretic description, rather than only a name and a citation to where they are defined?
  6. What is the current (as of this writing) status of any written, public, dated response by Scholze or Stix to Mochizuki's Sept. 2018 Cmt2018-08 — this file located none specifically targeting Cmt2018-08's (C12)-(C14) and (Lin) framing, only later, general restatements of their original 2018 position (e.g. Scholze's 2024 remark, row 13) that do not engage the (Lin)/"id-version" labels by name.
  7. LANA's team reports getting "stuck" at a point "closely related" to the SS-identified area (row 14) — related in what precise technical sense, and is it the same Step (xi) operation, a different step with similar structure, or something upstream of both? This file did not find a public, technical (as opposed to journalistic) description of LANA's specific obstruction; 06-lean-boundary.md is the place to check for any further detail on this specific question.

5.12 Explicit flags (do not silently resolve any of these)

  1. Pagination/version instability. SS's report cites Corollary 3.12 at "page 16" of IUT III; the hosted PDF used throughout this file (dated 2020-05-18 on Mochizuki's own publications list, i.e. about two years after SS wrote) has it at page 173. The theorem/corollary numbers appear stable; raw page numbers are not. This file cites by theorem/corollary number first, hosted-PDF page number second, and flags this gap explicitly rather than assuming the two versions are textually identical apart from pagination.
  2. Dating ambiguity on the SS report itself. The publicly-readable Bonn copy is self-dated "July 16, 2018" on its own title page; Mochizuki's hub page and his own Cmt2018-08 both call the same document the "August 2018" version/report. This file could not resolve which date is authoritative, or whether a mid-2018 revision explains the gap, and does not guess.
  3. Access-restriction change over time. The exact URL Quanta linked in Sept. 2018 for the SS report (.../~motizuki/SS2018-08.pdf, no protectedpdf- folder) returns HTTP 404 today; both of Mochizuki's currently-listed copies (protectedpdf-2018-05/, protectedpdf-2018-08/) return HTTP 403. The only publicly readable copy found by this research is the Bonn-hosted mirror (row 1). This is reported as an observed change in public accessibility, not as an inference about intent.
  4. Evidentiary asymmetry in secondary commentary, as detailed in §5.9: more independent commentary located by this research treats the objection as live/credible than treats it as rebutted, and this imbalance is flagged rather than treated as a tally of correctness.
  5. No direct, dated, public SS rebuttal of Cmt2018-08 was located. If later rounds of written exchange exist beyond what is listed in §5.1, they were not found by the searches performed for this file.
  6. One PDF's internal metadata date and title-page date disagree (Cmt2018-05: title page "July 2018," container metadata showing September 2018); noted but not pursued further, as it does not bear on the mathematical content.
  7. This guide's file set is assembled by several concurrent, independent research strands in one shared working directory, and that directory was observed to change, file-by-file, within minutes during this file's own drafting and review (e.g. guide/04-claim-dependencies.md, guide/03-iut-route.md, and guide/06-lean-boundary.md were each observed present, then absent, then present again — the last with revised wording — inside the single checking session on 2026-10-05 that produced this file). At the final check reported here, all of this file's cross-references — 00-map.md, 07-dictionary.md, 08-work-queue.md, and 06-lean-boundary.md — resolved, and 06-lean-boundary.md's own statement of its scope (that it does not itself evaluate the Scholze–Stix dispute) was checked against its then-current text, though §5.0 now states this only in paraphrase, not verbatim, to keep this file's opening section brief. None of this file's own factual claims about LANA depend on that sibling file's content regardless: every LANA-related claim here cites the New Scientist article (row 14) or the LANA GitHub README (row 16) directly. Given the demonstrated volatility, a reader opening this guide later should still treat every cross-reference link above as liable to have moved again since this paragraph was written, and should not assume the sibling files' exact wording is frozen.

5.13 Final disclaimer

This file reports what two identified parties said, in writing, on specific dates, at specific cited locations, about one specific step of one specific document — and what has and has not been said about it since. It is not a proof, not a refutation, not an endorsement of either side's characterization of the other's argument, and not a claim that the matter is settled by publication, by a prize, or by any formalization effort referenced above or in 06-lean-boundary.md. Where this research could not read a source, that is stated in §5.1 and §5.12, not silently filled in.