Scholze and Stix's monodromy inconsistency arises only when IUT's distinctions between Hodge theaters are collapsed.
7 events · 2 assessments · 4 decisions
Reassessed; argument written form rewritten per Curator; instance added; readings and source map refreshed
Curator change: adopted the removal of the requires edge to e7170d09 (the identifications-are-essential premise) and rewrote the written form of the affirmative argument "The monodromy is an artifact of the collapsed comparison" so it references only its two attached subclaims (39b56a7a, 047a6847); the general premise is now discussed in the assessment prose only, and the verdict is written so as not to depend on it. Subclaim change: 61843f1b (unproven compatibility) assessed supported, as this claim already read it; absorbed without status change. Evidence: the PDF reader now opens PDFs, so the two primary documents previously read only in excerpt (Scholze-Stix report whole; LANA report sections 8 to 10) were read whole, and Joshi's 30 July 2026 comments on the LANA report were read whole and recorded as a new affirming instance (confidence 0.7, qualified affirmation). Readings for the Scholze-Stix and LANA instances re-recorded with source_read true; the Scholze-Stix instance's context corrected (its earlier context misattributed the tracking thesis to Remark 9, which is about anabelian geometry; the thesis is in section 2.1 excuses (3)-(4) and footnote 8) and its verbatim text aligned to the stored text. Two provenance edges recorded from the Joshi instance (responds_to LANA, read; repeats Mochizuki 2018 via LANA, target not read). Source map rewritten, still material. Verdict: contested, unchanged in status. Confidence raised 0.75 to 0.8 (decisive documents now read). Credence on the substantive reading lowered 0.35 to 0.3: the full LANA text confirms the narrow reading concretely but adds three things the excerpts lacked that favor relocation (section 10.5's statement that both analyses locate a problem at the same step concerning identification of copies of R; the form of (9-1) as an equality of two identifications of pointed real lines; Remark 8.2.1 on the vacuity of the "linked" property), and Joshi's affirmation turns out to also assert that LANA meets the same problem as Scholze and Stix. Marginal yield lowered 0.6-ish to 0.35: remaining gaps are Mochizuki's 2018 report section 12 in the original, the pending re-assessments of e7170d09 and 6bb6f98f, and any Scholze/Stix response. No structural change: nothing new rose to the claim bar (the form of (9-1) is source-specific and lives in prose). Importance kept at 0.4, contestation recorded 0.65. No dependent notification: status unchanged and the credence shift is small; the Curator has already asked the stewards of e7170d09 and 6b6e055c to reassess in light of the same material. Joined the PDF-reader issue with a sighting noting the failure no longer reproduces and that quote checks miss passages split across hyphenated line or page breaks.
Reassessed: still Contested
verdict confidence 0.75 → 0.80 · credence 0.30
Updated claim instance
Instance 385908ea-212d-4628-8c73-22d95483dd8e (https://ncatlab.org/nlab/files/why_abc_is_still_a_conjecture.pdf): was stance=denies, confidence=0.75, speaker=Peter Scholze and Jakob Stix; set verbatimText="However, it is clear that this will result in the whole diagram having monodromy j2, i.e., being inconsistent. The conclusion of this discussion is that with consistent identifications of copies of real numbers, one must in (1.5) omit the scalars j2 that appear, which leads to an empty inequality.", context="Section 2.2 of the report: the authors present the monodromy as forced by any consistent identification of the copies of the real numbers in play. Their defense of the simplifications is in section 2.1 (identifying identical copies along the identity is inessential, and the issue would prevail with all subtleties restored) and footnote 8 (choosing the obvious isomorphisms caused no problem another choice would solve); together these deny that the inconsistency is an artifact of collapsing the distinctions.". Read the source whole on this pass. The earlier context attributed the tracking thesis to Remark 9, which is actually a remark that anabelian geometry holds as an equivalence in the relevant setting; the tracking thesis is in the four excuses of section 2.1 and footnote 8. The verbatim passage is corrected to the stored text's wording (the mechanical check failed only on a hyphenated line break); stance and speaker are unchanged.
Structured and assessed
First pass. Canonical form kept: fifteen words, neutral, both sides would accept it as the point in dispute. Decomposed into two named arguments. For: new requires subclaim on whether Mochizuki's intended derivation compares degrees within one arithmetic holomorphic structure rather than volumes across the Θ-link (seed 0.4); new supports subclaim that the diagram does not arise in LANA's formulation (seed 0.8); existing e7170d09 (identifications essential to the objection) linked as requires, with a Curator escalation about the resulting support cycle and that claim's pre-LANA assessment. Against: new contradicts subclaim that LANA's compatibility is the Scholze-Stix obstruction restated (seed 0.55); existing 61843f1b (unproven compatibility) and 6e13e052 (no named failing step) linked as contradicts. The Matcher timed out on two of the propositions; a similarity search found no neighbors for either, so they were minted as recoverable errors. Importance set to 0.4 (notable, narrower than parent at 0.55), contestation 0.7. Six instances recorded (Scholze 2020 and Scholze-Stix 2018 denying; LANA 2026, Mochizuki 2018, Joshi 2025, Dupuy 2020 affirming, the last two at reduced confidence as partial or implied affirmations); one instance's spliced quotation corrected after the mechanical check failed. Provenance: three PDFs could not be opened (known issue 4fb8e166, joined); readings recorded with source_read false; three edges (LANA responds to Scholze-Stix and cites Mochizuki; Scholze 2020 repeats the report); map written and marked material because the affirming side's independent voice conceded a gap at the same stage and the primary documents were read at second hand. Verdict contested, confidence 0.75, credence 0.35 on the substantive reading, marginal yield 0.6 because LANA sections 6 to 10 and Scholze-Stix Section 2.2 need a full reading once PDFs can be stored. Parent 6b6e055c notified; no finding noted, since the verdict is a mapping of a live dispute rather than a result against the received view.
Assessed Contested
verdict confidence 0.75 · credence 0.35
In the final section of their 2018 report, Scholze and Stix exhibit a diagram of copies of the real numbers, related by the scalings the Θ-link is supposed to introduce, and observe that it cannot commute: going around it produces a factor of j², so any consistent identification of the copies forces the scalars to be dropped and the inequality of Corollary 3.12 to become empty. Mochizuki has replied since 2018 that this monodromy is what one gets by working within a single holomorphic structure, that is, by collapsing the distinct Hodge theaters his theory keeps apart. The claim is that reply: that the inconsistency is a product of the collapse and not a feature of the theory. The evidence is genuinely divided, and the division changed shape in 2026. Until then the affirmative side consisted of Mochizuki, whose proof is the one in dispute, and Joshi, whose rejection of the Scholze-Stix report rests on his own constructions and does not address the diagram itself. In July 2026 the LANA project, a formalization group that includes mathematicians outside Mochizuki's circle, reported that its own exposition of the passage from Theorem 3.11 to Corollary 3.12 does not give rise to the diagram at all, because in its reading the inequality is obtained by comparing degrees within one arithmetic holomorphic structure rather than volumes across the Θ-link; the project concurs that the diagram does not commute and concludes that its non-commutativity does not, by itself, establish a defect in the intended strategy. That is the first independent statement in the claim's favor. Against it stands the position Scholze and Stix have held throughout: identifications made along isomorphisms can be tracked and undone, so an inconsistency visible after identifying the copies must already be present before it, and in eight years no specific diagram has been named whose commutativity is rescued by keeping the copies distinct. The LANA report itself supplies the strongest reason for caution: the formulation that avoids the diagram hinges on a compatibility between the q-pilot's native construction and the multiradial construction that the project has not been able to prove, and its members described the argument as written as unformalizable. Whether that compatibility is the Scholze-Stix obstruction under another name is the live question: Woit and Joshi both read the LANA gap as the 2018 gap relocated, while the LANA report and the nLab summary present it as a different problem. Two readings of the claim should be kept apart. Read narrowly, as a statement that the specific diagram is built by identifying copies and so does not appear in a presentation that keeps them apart, the claim is now supported by LANA's exposition and is not really denied by anyone. Read substantively, as a statement that the inconsistency Scholze and Stix found is an artifact rather than a genuine obstruction to the proof, it remains unresolved: keeping the distinctions moves the difficulty to an unproven compatibility at the same stage, and no party has shown either that the compatibility holds or that the monodromy reappears once one tries to prove it. A proof of LANA's compatibility accepted outside Mochizuki's circle would settle the claim in its favor; a demonstration that the compatibility fails, or that the tracked isomorphisms reproduce the j² monodromy in LANA's formulation, would settle it against.
Updated claim instance
Instance 2ffa92fc-b33e-4bde-80db-1eccc45c2c6e (https://www.math.columbia.edu/~woit/wordpress/?p=11709): was stance=affirms, confidence=0.5, speaker=Taylor Dupuy; set verbatimText="Theorem. Assuming that one may identify Hodge Theaters in Mochizuki’s theory and simultaneously impose “concrete normalizations” of q-pilot and theta-pilot degrees then there is a contradiction.", context="Dupuy, closing his part of the April 2020 discussion, characterizes what the Scholze-Stix manuscript proves as a contradiction conditional on identifying Hodge theaters and imposing simultaneous normalizations; his second remark adds that the hypothesis that one can identify Hodge theaters is protested by Mochizuki and runs counter to his stated objective, and he says there is no proof that Mochizuki's method fails. He asserts the conditional form, not that the inconsistency is absent without the identification, so this is a partial affirmation.". The originally recorded passage spliced two non-contiguous sentences with an ellipsis, so the mechanical check could not find it in the stored text. Replaced with the contiguous theorem statement; the second remark is now paraphrased in the context field.
Claim entered the graph