Finite lower-central coordinate systems with torsion accounted for

The images of a finite generating set generate $Q$. The torsion theorem makes $Q$ torsion-free nilpotent, so it has the integral central-refinement coordinates of F3. This invocation is made for $Q$, not for a possibly torsion-bearing $G$.

For either $E=G$ or $E=Q$, take a cyclic decomposition in each lower-central factor and one lift per generator. There are finitely many layers and finite lists. For $g\in E$, project to $E\mathrm{/}{\gamma }_2(E)$, obtain its unique cyclic-factor tuple, and let $p_1$ be the corresponding ordered lifted product. The remainder $p_1^{-1}g$ lies in ${\gamma }_2$. Repeat in ${\gamma }_2\mathrm{/}{\gamma }_3$, obtaining $p_2$; after layer $c$ the remainder is $1$. Thus $g=p_1\cdots p_c$ has the mixed ordered form.

Let $q:G\to Q$ be the quotient map. Induction gives $q({\gamma }_i(G))={\gamma }_i(Q)$: it is clear for $i=1$, and surjectivity sends the generators $[g,x]$ of the next term onto the generators $[q(g),q(x)]$. Hence $q$ induces a surjection ${\gamma }_i(G)\mathrm{/}{\gamma }_{i+1}(G)⟶{\gamma }_i(Q)\mathrm{/}{\gamma }_{i+1}(Q)\mathrm{.}$ If $x{\gamma }_{i+1}(G)$ is in its kernel, choose $y\in {\gamma }_{i+1}(G)$ with $q(x)=q(y)$. Then $t=x{y}^{-1}$ lies in $T(G)\cap {\gamma }_i(G)$ and $x{\gamma }_{i+1}(G)=t{\gamma }_{i+1}(G)$. Conversely every such $t$ lies in the kernel, so the kernel is the image of the finite set $T(G)\cap {\gamma }_i(G)$. The finite-kernel rank clause of [F4] therefore gives equal free ranks for the corresponding factors of $G$ and $Q$. Calling this common rank $r_i$, each construction in step 1.2 has exactly $r_i$ unbounded layer-$i$ exponents.

If two normalized products agree, project to $E\mathrm{/}{\gamma }_2$ to equate all first-layer coordinates, including the canonical residues. Their lifted first-layer products are then literally equal and may be cancelled on the left. Repeating in the next factor equates every coordinate. The identity has the all-zero tuple; for $E=1$ this is the empty tuple. Hence the mixed parametrization is bijective, whether or not some factors contain torsion.

If one instead lifts an integral coordinate representative $w$ of an element of $Q$ to $G$, the fiber consists exactly of the $\mathrm{\mid }T(G)\mathrm{\mid }$ elements $wt$ with $t\in T(G)$. Thus returning from the integral model retains a finite kernel representative. The mixed construction in steps 1.2–2.1 works directly in $G$ and does not discard these elements or identify the two coordinate systems.