Finite collection alphabets include commutators and torsion carries

For nonidentity $x$, define its depth as the largest $i\le c$ with $x\in {\gamma }_i$. Start with the given finite alphabet, the chosen coordinate lifts, and their inverses. Whenever two available letters $x,y$ have depths $i,j$, add $[x,y]$ and its inverse if nontrivial. Whenever a letter of depth $i$ has finite order $d$ in its factor, add $x^d$ and its inverse if nontrivial. Commutator outputs have depth at least $i+j$; carry outputs have depth at least $i+1$. Inversion preserves depth.

This closure is finite: regard every new letter as an expression built from initial letters by inverse, carry, and binary commutator operations, absorbing inverse into each operation so it is not an extra level. Along any branch of its expression tree, every non-inversion operation strictly increases depth. No branch has more than $c-1$ such operations. Binary trees of bounded height have bounded size; there are finitely many initial labels, and each carry exponent is uniquely determined by its input element. Induction on tree height therefore gives finitely many expressions and values. Closing under all these expressions yields the required alphabet. If $c=0$, it is empty after deleting identity.

Fix any resulting letter of depth $i$. Successive projection in the mixed coordinates writes it as a fixed word in the weight-$i$ coordinate lifts followed by a word of weights at least $i+1$: all earlier coordinates vanish by the uniqueness construction. There are finitely many letters, so the lengths of these replacements have a common finite bound. Replacement of an inverse uses the reversed inverse word, with the same bound. Conversely fixed coordinate lifts also have finite words in any alphabet generating their lower-central term. These words account for finite changes of layer alphabets without treating redundant letters as independent coordinates.