Root correction criterion for uniformity ($u\mapsto u^p:U_j/U_{j+1}⟶U_{j+1}/U_{j+2}$)

For odd $p$ and a finitely generated powerful pro-p group, put $U_j=G^{p^j}$. The displayed group homomorphisms are surjective. If one has a nonzero kernel, choose $x\in U_j\setminus U_{j+1}$ with $x^p\in U_{j+2}$. Successive corrections $x\mapsto x{u}_r$, with $u_r\in U_r$, cancel its $p$-th power one layer at a time, using $(x{u}_r{)}^p\equiv x^p{u}_r^p(\mathrm{mod}{U}_{r+2})$. Compactness and the separated power filtration give a nonidentity limit of order $p$. Thus a torsion-free group has constant layer indices and is a uniform pro-p group. Conversely constant indices make every displayed map injective; comparison at the first differing power layer proves that $x\mapsto x^p$ is injective, so there is no nontrivial torsion.