of (real) EL numbers will be approximated by Cauchy-like distribution .

Let us assume statistical distribution of real EL constants has PDF $P(x)$ empirically derived from histogram in Fig. \[StatReals\] or is given by . Let the target number $z$ is know with precision $\sigma$, and statistical distribution of error for $z \pm \sigma$ is $Q(x, z, \sigma)$. $Q$ might be normal or uniform distribution, for example. Then conditional probability $p_k$, that $k$-th tested formula **is not random** match for target value $z \pm \sigma$ is: $$\begin{aligned}
p_k =  \prod_{i=1}^k   \left( 1-  \frac{\int_{-\infty}^{\infty} P(\xi) Q(\xi) \; d \xi}{Q(z)} \right) = \\ \nonumber
=
\left( 1-  \frac{\int_{-\infty}^{\infty} P(\xi) Q(\xi) \; d \xi}{Q(z)} \right)^k.\end{aligned}$$

Then, replacing $P(\xi) \simeq P(z)$ and integrating, likelihood $\mathcal{L}$ for proper identification of target constant $z$ with i-th unique value $x_i$ is: $$\label{Id_likelihood}
\mathcal{L} = \left[ 1-   \frac{P(z)}{Q(z)} \right]^{k_3} Q(x_i, z, \sigma),$$ where $i=k_3$, i.e. number of unique values tested so far. To understand , it might be approximated by $$\label{Id_likelihood2}
\left( 1- \kappa \, \sigma \; P(z) \; k_3 \right) Q(x_i, z, \sigma),$$ assuming $\sigma \ll z$ and using Maclaurin expansion $(1+\alpha)^k \simeq 1 + \alpha k + \ldots$. Value of $\kappa$ depends on error distribution $Q$, 