= Derived-series growth of the Nottingham group
{title2=$\log_p[\mathcal N:\mathcal N^{(n)}]=O(\lambda^n),\quad\lambda<3$}
For odd $p$, <group commutator> collection in the depth filtration gives $[\mathcal N_m,\mathcal N_m]=\mathcal N_{2m+1}$. Every degree $k\ge2m+1$ occurs as a leading <group commutator> degree by choosing $r,s\ge m$, $r+s=k$, and $r\not\equiv s\pmod p$; changing $r$ by one changes $r-s$ by two. Closed successive coefficient correction then fills the tail. Therefore $\mathcal N^{(n)}=\mathcal N_{2^{n+1}-1}$ and $\log_p[\mathcal N:\mathcal N^{(n)}]=2^{n+1}-2$. In characteristic two the sharper collection theorem gives $\log_2[\mathcal N:\mathcal N^{(n)}]\le C\kappa^n$ for a constant $C$, with $\kappa=(1+\sqrt{17})/2<3$. In both cases the exponential base is strictly smaller than three.
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