Joshi's precisification argument
The argument
Because Mochizuki's IUT papers never define arithmetic holomorphic structure precisely enough to prove that two such structures differ, the notion IUT relies on is fixed only by the role it must play; and because Mochizuki's Key Principle of Inter-Universality demands arbitrary geometric base-points that only algebraically closed perfectoid fields supply, structures indexed by untilts of such fields are the only precise notion that can fill that role, so Joshi's structures are the structures IUT was implicitly about. That Joshi's theory yields a proof of Mochizuki's Corollary 3.12 would corroborate the identification by showing his structures doing the work IUT assigns to its own.
Does it work?
Not yet evaluated.
Premises
- 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 →Mochizuki's IUT papers give no definition of arithmetic holomorphic structure precise enough to prove that two such structures differ.
- 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 →Mochizuki's Key Principle of Inter-Universality requires arbitrary geometric base-points for tempered fundamental groups, which only algebraically closed perfectoid fields supply.
- 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 →Joshi's arithmetic Teichmüller theory yields a valid proof of Mochizuki's Corollary 3.12.
Where it is made
Passages where sources make, criticize or mention this argument. They record where it appears, not how strong it is, and none has precedence over another.
No source is recorded as making it.
Earlier ratings
- Sep 18, 2026
Granting its premises the inference reaches only a reconstruction: if IUT's notion is fixed by its role and only untilt-indexed structures can fill that role, Joshi's structures are the best available precisification, which is weaker than their being the same objects. The argument lives or dies on the reading of the Key Principle as demanding base-points that only perfectoid fields supply, which Mochizuki rejects and which is asserted without an argument for its exclusivity; the imprecision of Mochizuki's definition is plausible but cuts both ways, since an underspecified target cannot be shown identical to anything. Joshi's claimed proof of Corollary 3.12 would corroborate the argument only if it carried Mochizuki's global content, which Scholze and Mochizuki dispute.