In his latest 115-page overview, Mochizuki spends some time explaining "alien copies" by the analogue of evaluating the Gaussian integral by squaring it and introducing a second variable/dimension. In sections 1.1 to 1.7 (pages 5-10) he goes through this, and then in section 3.8 (pages 89-94) he revisits the analogy.

I have two questions about this. Firstly, one gets that $I^2=(\int_{-\infty}^\infty e^{-x^2}\,dx)^2=\pi$, and by positivity one finds $I=\sqrt\pi$. What is the analogue in the IUT setting of "positivity" here? Do the choices up to a unit not matter somehow?

Secondly, I wonder about the scope of alien copies. At the end of the day, if I read 3.8 correctly in #10 and #11 in the table on page 92 and the computations in 3.7(iv) (page 89), the main result of the analogy is to evaluate (to first order) the **left side** of the inequality at the top of page 87.

So my question is: are alien copies "only" useful for this Gaussian-like evaluation, or do they play a role throughout? (NB: the word "alien" does not appear in the text from pages 22-90.) Alternatively, do they also enlighten the derivation of the inequality (see 3.7(ii) starting on page 85) in the first place, or the estimation of its right side?

Since no one has answered yet, maybe I'll add a third question. In 2.2 on page 11, a distinction (quite important I gather) between étale-like and Frobenius-like structures is made, the former being "zero mass" objects, such as (1.4) the differentials "$dx\,dy$" in the Gaussian evaluation. Are there Frobenius-like structures with the Gaussian analogue, and if so, what are they?

everythingin abc-conjecture can be considered number-theory, no? (PS: I did know this recommendation, but carefully read the "hints for tags", and didn't see it there) $\endgroup$1more comment