>>31
つづき

I may have not expressed this clearly enough in my manuscript with Stix, but there is just no way that anything like what Mochizuki does can work. (I would not make this claim as strong as I am making it if I had not discussed this for with Mochizuki in Kyoto for a whole week; the following point is extremely basic, and Mochizuki could not convince me that one dot of it is misguided, during that whole week.) It strikes deep into my heart to think that in the name of pure mathematics, an institute could be founded for research on such questions, and I sincerely hope that this will not come back to haunt pure mathematics.

The reason it cannot work is a theorem of Mochizuki himself. This states that a hyperbolic curve X over a p-adic field K (maybe with some assumptions, all of which are always satisfied in all cases relevant to IUT) is determined up to isomorphism by its fundamental group π1(X),
and in fact automorphisms of X are bijective with outer automorphisms of π1(X).
Thus, the data of X is completely equivalent to the data of π1(X) as a profinite group up to conjugation.
In IUT, Mochizuki always considers the latter type of data, but of course up to equivalence of groupoids this makes no difference. (The passage back and forth is even constructive, by another result of Mochizuki.)

Mochizuki claims that by replacing X by π1(X),
things can happen that cannot otherwise happen. Examples are given concerning the action of π1(X)
on certain associated monoids. We discussed this at very great length in Kyoto, but none of these examples carried any actual content. Note that any potential non-commutativity of some diagram that results from identifying π1(X)’s
via isomorphisms of X’s could not possibly be resolved by using some other isomorphism of π1(X)’s ? all of them come from isomorphisms of X’s!

つづく