Question on derivation of $\omega$ in surreal numbers

Question

This paper gives a derivation definition on log-atomic surreal numbers:

where the logarithm with lower index means iterated logarithm.

I think — I may be wrong — that $\omega$ is a log-atomic number. In that case, the derivation follows this formula.

But we know that the derivation of $\omega$ should be $1$, so, this long expansion should be in the limit equal to 1 (the sum should be zero). But it is not.

What's wrong here?

Note that the first iteration looks indeed as $1$ if $\omega$ is inserted for $\lambda$.

Answer 1

Note that $\partial^\prime_{\mathbb L}$ (with $\prime$) is mentioned there as a first natural definition it may come to mind, of a derivation satisfying the properties in Proposition 6.5 , but it is observed few lines below that it doesn’t have the nice properties one would like, in particular:

no $x$ satisfies $\partial^\prime_{\mathbb L}(x)=1.$

A derivation $\partial_{\mathbb L} $ (without $\prime$) for which in particular $\partial_{\mathbb L}(\omega)=1$ is introduced immediately after, and it is proved that it is the simplest in a suitable sense.