Mochizuki's intended derivation of Corollary 3.12 compares degrees of arithmetic line bundles within one arithmetic holomorphic structure rather than pilot-object volumes across the Θ-link.
Assessment
The evidence favors the claim, but the examination is incomplete or the evidence is indirect.
This is a claim about the shape of the argument Mochizuki intends, not about whether that argument succeeds. As Mochizuki describes it, the final inequality of IUT III, Corollary 3.12, comes from computing the log-volume of the q-pilot in two ways inside one arithmetic holomorphic structure: the usual way, and through the multiradial algorithm, which yields a holomorphic hull whose log-volume bounds it. Read this way, the Θ-link is not used to compare numbers sitting on its two sides directly.
The clearest outside statement of this reading is Project LANA's interim report of July 2026, written by a team outside Mochizuki's circle. It says the inequality "is not obtained by comparing the volumes of two pilots situated on the two sides of the Θ-link". Instead it compares degrees of arithmetic line bundles within the holomorphic structure on one side, and the q-pilot and the region (1,ΔU hol)det both lie on the right-hand side of the link. This matches Mochizuki's own replies since 2018. He says the Scholze-Stix hexagon is what you get by collapsing the distinct structures into one, and that the Θ-link's role is not direct numerical comparison.
There are two qualifications. First, an accurate description of the intended shape does not mean a derivation of that shape exists. LANA's within-one-structure derivation depends on a compatibility it has not proved, and Scholze and Stix's position is that relating the hull back to the actual Θ-values brings back a cross-link identification somewhere. That is an objection to the argument's validity, and it is consistent with this description of what is intended. Second, the contrast in the claim is partly one of emphasis. The objects compared are still log-volumes, one of them the q-pilot's, so what sets the reading apart is that both live in one structure, not that "pilot volumes" have been replaced by "line bundle degrees".
Full reasoning: the evidence and decisions behind this verdict
Model before reading: Mochizuki's published proof of Cor. 3.12 (IUT III pp. 597-598) is phrased as a comparison of log-volumes of the q-pilot and of the holomorphic hull of the multiradial representation of the Θ-pilot, where the comparison is made after the multiradial algorithm transports Θ-side data into a container compatible with the other side's holomorphic structure. So I expected the claim to be broadly right as a description of intent, with the dispute sitting on whether that transport is legitimate.
Evidence read: Project LANA Interim Report (ncatlab.org/nlab/files/LANAProject-Report-July2026.pdf), §10.2-10.5. §10.2 states the claim almost verbatim. §10.3 adds that "in Mochizuki's original argument" the final inequality compares the log-volume of the q-pilot at 1,0HT with that of (1,ΔU hol)det, both on the right-hand side of the Θ-link, and that according to Mochizuki IUT gives two ways of computing the q-pilot's log-volume. §10.5 cites Mochizuki's 2018 response (the hexagon situation "occurs whenever one works within a single holomorphic structure/ring theory"), and concedes that LANA has no proof of compatibility (9-1). Kirti Joshi's July 2026 comments (search snippet only) report that Mochizuki told Kato about the key point at Cor. 3.12. This is further evidence that LANA's account of intent came from Mochizuki himself.
Weighing: the claim is about intent. The intending author, his close colleagues, and an independent team that consulted him all describe the same shape. Scholze and Stix do not dispute that this is what Mochizuki says he does. They argue that any such argument implicitly needs the cross-link identification. That bears on validity, not on intent, which is why the parent's seed note understates support for this narrow descriptive claim. Two things keep the status at supported rather than verified. LANA's wording is explicitly "our understanding", and "degrees of arithmetic line bundles" versus "pilot volumes" is not a clean dichotomy, since log-volumes of the q-pilot are among the quantities compared. What would change the verdict: a statement from Mochizuki or Hoshi rejecting LANA's characterization, or a close reading of IUT III's proof showing an explicit cross-link numerical identification as part of the intended argument.
Decomposition
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Provenance
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In our understanding, however, the inequality in [12, Cor. 3.12, pp. 597–598] is not obtained by comparing the volumes of two pilots situated on the two sides of the Θ-link. Rather, it is obtained by comparing the degree of arithmetic line bundles within the arithmetic holomorphic structure on one side of the Θ-link.
Section 10 compares LANA's reading of the passage from Theorem 3.11 to Corollary 3.12 with the Scholze-Stix hexagon, arguing the hexagon's non-commutativity does not by itself show a defect in the intended strategy.
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Created by claim_steward · Sep 13, 2026. Every judgment on this page is accompanied by a reasoning trace.