\begin{align} H &= \frac{1}{2} \ln (2\pi\sigma^2) + \frac{1}{2}\\ &= \frac{1}{2} \ln (2\pi e \sigma^2) \end{align} is the analytical solution for the entropy of a Gaussian random variable, such as a returns series, in the textbook Financial Machine Learning.
The author then re-works one of the above into a formula for entropy-implied volatility by isolating $\sigma$ by itself, and renaming it $\sigma_H$:
$$\sigma_{H} = \frac{e^{H} - \frac{1}{2}}{\sqrt{2\pi}}$$
but isn't this useless because, in order to calculate $H$, you need sample volatility $\sigma$ anyway? In other words, to calculate $\sigma_H$, you need $H$, which itself in turn requires $\sigma$, meaning $\sigma = \sigma_H$. It's fair enough that $H$ is a function of $\sigma$, but $\sigma_H$ is a function of $H$ as well as its own self, $\sigma$
Is the $\sigma_H$ formula just a small exercise in algebra that is circularly redundant given that it itself ($\sigma$) is required as an input?
