[Quasi-convergence of enumerated formulas and constant recognition \[Criterion1\] ]{}

Enumerated formulas provide sequence of elementary transcendental $\exp-\log$ (EL) numbers. For they are, in order of increasing Kolmogorov complexity $K$: $$\begin{aligned}
\label{ELseq}
K=1&\qquad e,                    \\
K=2&\qquad \text{none},   \nonumber        \\ 
K=3&\qquad 1, e^e,        \nonumber         \\
K=4&\qquad \text{none},    \nonumber         \\
K=5&\qquad 0, 1/e, e^{e^e}, e^{e^2}, \nonumber \\
\ldots  \nonumber\end{aligned}$$ where repeated numbers were omitted. Position within the same level of $K$ is unspecified, but follows from presumed enumeration loop and base-$n$ digits association with buttons. From sequence we can extract subsequence of progressively better approximations for target number $z$, e.g., using example from the introduction, $z=1.8226346549662422143937682155941\ldots$. Analysis of sequences obtained this way is main goal of current section. We are interested in convergence properties and criteria for termination of sequence.

Let us assume for the moment, that we are able to provide on demand as many decimal digits for target number $z$ as required. There are two mutually exclusive possibilities: (i) number $z$ is in $\exp-\log$ class, defined e.g. by or , or (ii) $z$ is true non-elementary transcendental constant outside EL class. In the former case, we expect that error eventually, at some finite $K$, will drop to an infinitesimally small value, 