= Solution
We next bound the closed <derived series>. For odd $p$, the depth calculation yields
$$
[\mathcal N_m,\mathcal N_m]=\mathcal N_{2m+1}.
$$
For the inclusion from left to right, two series of depth at least $m$ commute modulo $\mathcal N_{2m+1}$: at depth $2m$ the only possible leading pair has equal depths and its coefficient vanishes. For the reverse inclusion, every depth $k\ge2m+1$ can be written $r+s=k$ with $r,s\ge m$ and $r\not\equiv s\pmod p$. Start with $r=m,s=k-m$; if their difference vanishes modulo $p$, replace them by $r=m+1,s=k-m-1$, which is allowed except at $k=2m+1$, where the original difference is already nonzero. The difference changes by $2$, nonzero for odd $p$. <Commutators> thus supply every high <leading coefficient>, and closed successive coefficient correction fills the tail.
With $\mathcal N^{(0)}=\mathcal N_1$, induction on $n$ gives
$$
\mathcal N^{(n)}=\mathcal N_{2^{n+1}-1},\qquad\log_p[\mathcal N:\mathcal N^{(n)}]=2^{n+1}-2.
$$
For $p=2$ use the standard characteristic-two derived-series collection bound, explicitly stated as an additional result:
$$
\log_2[\mathcal N:\mathcal N^{(n)}]\le C\kappa^n,\qquad\kappa=\frac{1+\sqrt{17}}2<3,
$$
for a fixed constant $C$. One can take a fixed enlarged constant from the tail estimate $\mathcal N^{(n)}\supseteq\mathcal N_{\lceil4\kappa^n\rceil+1}$. The more involved two-coefficient collection is necessary in characteristic two; the exact odd-prime tail formula is not asserted there.
In either case there are constants $C'>0$ and $\lambda<3$ such that $\log_p[\mathcal N:\mathcal N^{(n)}]\le C'\lambda^n$. If an analytic structure existed, its <dimension> $d$ would be positive, since a zero-dimensional analytic <pro-p group> is discrete and <compact>, hence finite. The second alternative of the supplied theorem would require
$$
\log_p[\mathcal N:\mathcal N^{(n)}]\ge d\,3^{n-s}
$$
for all sufficiently large $n$, for some fixed $s$. Combining the bounds gives $d3^{-s}\le C'(\lambda/3)^n$, impossible as $n\to\infty$.
The root solution proved hereditary just infiniteness, so the supplied theorem applies. Part (i) excluded its linear alternative, and this calculation excludes its growth alternative. Hence \b[the <Nottingham group> is not analytic over any pro-$p$ ring].
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