Over a commutative $\mathbb{Q}$-algebra, $\mathrm{CYC}(\mathcal{A})$ has generating function ${\sum }_{k\ge 1}\frac{\phi (k)}{k}\log \frac{1}{1-A(x^k)}$

For each $m\ge 1$, let ${\mathcal{C}}_m$ be the class of cycles of length $m$. Since every object of $\mathcal{A}$ has positive size, an $m$-tuple of total size $n$ can use only entries of size at most $n$, so every size layer of ${\mathcal{A}}^m$ is finite. Applying [L1] degree by degree to the cyclic action of $C_m$ therefore gives $\mathrm{OGF}({\mathcal{C}}_m)=(1\mathrm{/}m){\sum }_{r=0}^{m-1}{F}_{m,r}(x)$, where $F_{m,r}(x)$ is the generating function of the $m$-tuples fixed by rotation by $r$ places.

Put $d:=\gcd (m,r)$ and $k:=m\mathrm{/}d$. By [L2], a tuple fixed by rotation by $r$ is obtained by repeating one block of length $d$. Each entry in that block is counted $k$ times in the full cycle, so the generating function of such fixed tuples is $F_{m,r}(x)=A(x^k{)}^d=A(x^{m\mathrm{/}d}{)}^d$.

Fix a divisor $k$ of $m$, and write $d=m\mathrm{/}k$. The rotations with $m\mathrm{/}\gcd (m,r)=k$ are exactly the integers $r=ds$ with $0\le s<k$ and $\gcd (s,k)=1$, so [L3] shows that there are $\phi (k)$ of them. Step 1.2 therefore gives $\mathrm{OGF}({\mathcal{C}}_m)=(1\mathrm{/}m){\sum }_{k\mid m}\phi (k)A(x^k{)}^{m\mathrm{/}k}$.

Summing step 2.1 over all $m\ge 1$ and writing $m=dk$ yields $\mathrm{OGF}(\mathrm{CYC}(\mathcal{A}))={\sum }_{k\ge 1}(\phi (k)\mathrm{/}k){\sum }_{d\ge 1}A(x^k{)}^d\mathrm{/}d$. Because $A(0)=0$, every degree receives contributions from only finitely many pairs $(k,d)$, so this regrouping is coefficientwise finite.

Applying [L4] with $u=-A(x^k)$ gives ${\sum }_{d\ge 1}A(x^k{)}^d\mathrm{/}d=\log (1\mathrm{/}(1-A(x^k)))$. Substituting this into step 3.1 gives the stated cycle formula.