trailKirti Joshi's arithmetic Teichmüller theory establishes the existence of distinct arithmetic holomorphic structures.
The distinctions IUT draws between isomorphic copies of objects carry essential mathematical content.
The simplifying identifications in Scholze and Stix's analysis are essential to their objection to Corollary 3.12.
Existence via untilts· for
A perfectoid field of characteristic p has infinitely many distinct untilts to characteristic zero, parametrized by the Fargues–Fontaine curve.
atomic
Joshi's arithmetic Teichmüller theory yields a valid proof of Mochizuki's Corollary 3.12.
atomic
Definitional mismatch with IUT· against
Joshi's arithmetic holomorphic structures are the same objects as those of Mochizuki's inter-universal Teichmüller theory.
Mochizuki's IUT papers give no definition of arithmetic holomorphic structure precise enough to prove that two such structures differ.
Mochizuki's Key Principle of Inter-Universality requires arbitrary geometric base-points for tempered fundamental groups, which only algebraically closed perfectoid fields supply.
In IUT, "distinct arithmetic holomorphic structures" means ring structures related by a non-ring-theoretic link, not non-isomorphic structures.
The untilt data that defines Joshi's arithmetic holomorphic structures plays no role in the constructions of Mochizuki's IUT papers.
claim · empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →claim page ↗︎
Kirti Joshi's arithmetic Teichmüller theory establishes the existence of distinct arithmetic holomorphic structures.
⇄ContestedCredible evidence or argument exists on multiple sides.constitution →
2depended on by
this rests on ↓
claimA box is a claim: a single proposition the graph assesses, with its own page and map. Click any claim to centre the map on it.constitution →argumentA pill is an argument: one line of reasoning stating how the claims beneath it combine to bear on the claim above it, for or against. Arguments are not destinations; click their claims to explore.constitution →✓verifiedThe claim traces to reliable primary sources through a clear chain of evidence.constitution →↑supportedEvidence favors the claim, but the chain is incomplete or the sources are secondary.constitution →⇄contestedCredible evidence or argument exists on multiple sides.constitution →○unsupportedNo credible evidence found, though the claim is not contradicted.constitution →✕contradictedAvailable evidence weighs against the claim.constitution →?unknownInsufficient information to assess.constitution →◌unassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →verified factopen questionvalue premisetheoremprizeA double ring marks a claim with a live prize: an amount in owls offered for a machine-checked proof or disproof of its published formal statement. The amount is in the preview and on the claim page; a prize changes nothing about how the claim is assessed or how important it is judged to be.prize rules →⊢machine-checked⊢ marks a claim whose published formal statement has a machine-checked proof or disproof. The checker confirms the proof; the verdict beside it is still the steward's judgment of the claim as worded.prize rules →supportsthis provides evidence for the parentsteward instructions →contradictsthis argues against the parentsteward instructions →assumesbackground the parent's framing takes as givensteward instructions →requiresa load-bearing premise: the parent is false without itsteward instructions →Fig. Detail falls off with distance; every claim is an address.