In Chapter 8 of Nassim Nicholas Taleb's Statistical Consequences of Fat Tails, Taleb defines a metric $\kappa$. The relevant definitions are as follows: $$\mathbb{M}(n)=\mathbb{E}\left(\left\lvert\sum_{i=1}^n X_i-\mathbb{E}\sum_{i=1}^n X_i\right\rvert\right)$$ $$\kappa_{{n_0},n}=2-\frac{log(n)-log(n_0)}{log\left(\frac{\mathbb{M}(n)}{\mathbb{M}(n_0)}\right)}$$ where the $X_i$ are i.i.d. variates from some distribution with a mean. Thus $\mathbb{M}$ is the MAD for $n$ summands and $\kappa$ is a measure of the "rate of convergence" of $\mathbb{M}$. Solving the $\kappa$ definition for $\mathbb{M}(n)/\mathbb{M}(n_0)$ yields: $$\frac{\mathbb{M}(n)}{\mathbb{M}{n_0}} = \left(\frac{n}{n_0}\right)^{\frac{1}{2-\kappa_{n_0,n}}}$$
Now, let $G$ be a Guassian distribution and $V$ an arbitrary distribution with a mean such that $\mathbb{M_g}(1)=\mathbb{M_v}(1)$. For a given $n_g > 0$, Taleb defines the corresponding $n_v$ to be the smallest value such that: $$\mathbb{E}\left(\left\lvert\sum_{i=1}^{n_v}\frac{X_{v,i}-m_v}{n_v}\right\rvert\right) \le \mathbb{E}\left(\left\lvert\sum_{i=1}^{n_g}\frac{X_{g,i}-m_g}{n_g}\right\rvert\right)$$ i.e., the smallest value where the MAD of the summands is no greater than $n_g$ summands of the Gaussian.
Taleb then claims that: $$n_v=n_g^{-\frac{1}{\kappa_{1,n_g}-1}}$$ I don't follow that claim. Beginning with the inequality, I reason as follows: $$\frac{1}{n_v}\mathbb{M_v}(n_v) <= \frac{1}{n_g}\mathbb{M_g}(n_g) = \frac{1}{n_g}\sqrt{n_g}\mathbb{M_g}(1)=\frac{1}{\sqrt{n_g}}\mathbb{M_v}(1)$$ because the distributions have the same scale. So: $$\frac{1}{n_v}\frac{\mathbb{M_v}(n_v)}{\mathbb{M_v}(1)}\le\frac{1}{\sqrt{n_g}}$$ $$\frac{1}{n_v}n_v^\frac{1}{2-\kappa_{1,n_v}}\le\frac{1}{\sqrt{n_g}}$$ I can solve this inequality for $n_v$, but it does not yield the value given by Taleb. What am I missing?