In surreal numbers, the log-atomic numbers are those numbers that can be obtained from $\omega$ and its powers via iterated logarithm or exponential function.
At the same time, Timothy Chow's EL-numbers are those numbers which can be obtained from $0$ using field operations, logarithmic and exponential functions, including iterated logarithms or exponentials.
So, the two concepts are quite similar, but for log-atomic numbers the starting point is $\omega$, but for EL-numbers, the starting point is $0$.
Now, from Laplace transform or Harmonic series it follows that these two expressions with divergent integrals are equal: $\int_0^1 \frac 1xdx-\gamma=\int_1^\infty \frac1x dx$.
The later integral can be assumed to be eqivalent of the germ of the function $\ln x$ at infinity (if we postulate that germs corresponding to purely-infinite surreal numbers are equivalent to the corresponding improper divergent integrals), and given the canonical H-field structure on surreals, is equal to $\ln \omega$.
The first integral, on the other hand, can be seen as the germ of $\ln x$ at zero, so it looks natural that if we to define logarithm of zero, it should be equal to $\ln 0=\int_0^1 \frac 1xdx=-\ln \omega -\gamma$.
Extending the definition of EL-numbers to surreals, it follows this way that $e^\gamma\omega=\exp(-\ln 0)$, an EL-number. But it is not a log-atomic number.
On the other hand, $\omega$ is a log-atomic number, but seemingly is not an EL-number.
I wonder, why it is the case then? Can these definitions be reconcilled, and $\omega$ somehow established as an EL-number?
It seems, $\omega$ would be an EL-number if the constant $e^\gamma$ is an EL-number as well. And we know that $e^{-\gamma}$ and $\frac4\pi$ demonstrate surprising parallels, the later constant being an EL-number...
