= BCH convergence on a powerful Lie lattice
{c}
{title2=$x*y=\operatorname{BCH}(x,y)$}
The <Baker--Campbell--Hausdorff formula> converges integrally on a <powerful Lie lattice>. A degree-$m$ Lie term has bracket divisibility $p^{\epsilon(m-1)}$; the standard integral coefficient bound loses at most $\lfloor(m-1)/(p-1)\rfloor$ powers of $p$. The remaining valuation tends to infinity. Associativity follows from the formal exponential identity, the identity is zero, the inverse is $-x$, and $x^{*n}=nx$. The resulting <pro-p group> is a <uniform pro-p group>, with power subgroups $p^nL$.
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