[proofplan] We expand the mixed moment by the free moment-cumulant formula as a finite sum over noncrossing partitions. Freeness implies that a cumulant factor vanishes whenever its block contains entries from more than one free subalgebra. Therefore only colour-preserving noncrossing partitions contribute, and each surviving factor is a cumulant computed entirely inside one marginal subalgebra. The desired polynomial is precisely this finite colour-preserving cumulant expansion. [/proofplan] [step:Introduce the colour pattern and ordered block tuples] Let $c:\{1,\dots,n\}\to I$ be the colour map defined by $c(j)=i_j$, and let $NC(n)$ denote the finite set of noncrossing partitions of $\{1,\dots,n\}$. For a block $B\subset \{1,\dots,n\}$ with $B=\{b_1<\cdots<b_m\}$, define the ordered tuple $a_B \in \mathcal A^m$ by \begin{align*} a_B=(a_{b_1},\dots,a_{b_m}). \end{align*} A block $B$ has constant colour if $c(b_1)=\cdots=c(b_m)$. Define $NC_c(n)$ to be the set of noncrossing partitions $\pi \in NC(n)$ such that every block $B \in \pi$ has constant colour. For each $m \in \mathbb N$, let $\kappa_m:\mathcal A^m\to \mathbb C$ denote the $m$-th free cumulant associated to $\varphi$. For a noncrossing partition $\pi \in NC(n)$, define its partition cumulant on $(a_1,\dots,a_n)$ by \begin{align*} \kappa_\pi(a_1,\dots,a_n)=\prod_{B\in \pi}\kappa_{|B|}(a_B). \end{align*} This product is finite because every partition of $\{1,\dots,n\}$ has finitely many blocks. [/step] [step:Expand the moment by the free moment-cumulant formula] By the free [moment-cumulant formula for free cumulants](/theorems/7107), applied to the tuple $(a_1,\dots,a_n)\in\mathcal A^n$, \begin{align*} \varphi(a_1\cdots a_n)=\sum_{\pi\in NC(n)}\kappa_\pi(a_1,\dots,a_n). \end{align*} Substituting the definition of $\kappa_\pi$ gives \begin{align*} \varphi(a_1\cdots a_n)=\sum_{\pi\in NC(n)}\prod_{B\in \pi}\kappa_{|B|}(a_B). \end{align*} The sum is finite because $NC(n)$ is a finite set. [guided] The first move is to replace a moment by cumulants. The free moment-cumulant formula states that, for every $n \in \mathbb N$ and every tuple $(x_1,\dots,x_n)\in \mathcal A^n$, the moment $\varphi(x_1\cdots x_n)$ equals the sum over all noncrossing partitions of $\{1,\dots,n\}$ of the product of cumulants over the blocks. We apply this formula to the specific tuple $(a_1,\dots,a_n)\in \mathcal A^n$. For each partition $\pi\in NC(n)$, the contribution is \begin{align*} \kappa_\pi(a_1,\dots,a_n)=\prod_{B\in \pi}\kappa_{|B|}(a_B), \end{align*} where $a_B$ is the ordered tuple of entries whose indices lie in $B$. Therefore \begin{align*} \varphi(a_1\cdots a_n)=\sum_{\pi\in NC(n)}\prod_{B\in \pi}\kappa_{|B|}(a_B). \end{align*} This is already a finite polynomial expression in cumulants of subtuples of $(a_1,\dots,a_n)$, because the set $NC(n)$ is finite and each summand is a finite product. The remaining point is to identify which of these cumulants are marginal cumulants and which vanish because of freeness. [/guided] [/step] [step:Discard every partition containing a mixed-colour block] Let $\pi\in NC(n)$. Suppose that $\pi$ has a block $B=\{b_1<\cdots<b_m\}$ that does not have constant colour. Then the tuple $a_B=(a_{b_1},\dots,a_{b_m})$ contains entries from at least two distinct free subalgebras. By the mixed-cumulant characterization of freeness, applied to the free family $(\mathcal A_i)_{i\in I}$ and to the tuple $a_B$, mixed free cumulants vanish. Hence \begin{align*} \kappa_{|B|}(a_B)=0. \end{align*} Therefore \begin{align*} \kappa_\pi(a_1,\dots,a_n)=\prod_{C\in \pi}\kappa_{|C|}(a_C)=0. \end{align*} It follows that only partitions in $NC_c(n)$ contribute: \begin{align*} \varphi(a_1\cdots a_n)=\sum_{\pi\in NC_c(n)}\prod_{B\in \pi}\kappa_{|B|}(a_B). \end{align*} [guided] Now we use freeness to remove all terms that cannot contribute. Take a noncrossing partition $\pi\in NC(n)$. If every block of $\pi$ has constant colour, then $\pi$ belongs to $NC_c(n)$ and we keep it. Suppose instead that some block $B=\{b_1<\cdots<b_m\}$ contains at least two colours. This means there exist $r,s\in\{1,\dots,m\}$ such that $i_{b_r}\ne i_{b_s}$. Since $a_{b_r}\in \mathcal A_{i_{b_r}}$ and $a_{b_s}\in \mathcal A_{i_{b_s}}$, the ordered tuple \begin{align*} a_B=(a_{b_1},\dots,a_{b_m}) \end{align*} contains variables from at least two different free subalgebras. The mixed-cumulant characterization of freeness says that any free cumulant whose arguments come from more than one algebra in a free family is zero. Applying that characterization to the tuple $a_B$ gives \begin{align*} \kappa_{|B|}(a_B)=0. \end{align*} But the partition contribution $\kappa_\pi(a_1,\dots,a_n)$ is a product over all blocks of $\pi$: \begin{align*} \kappa_\pi(a_1,\dots,a_n)=\prod_{C\in \pi}\kappa_{|C|}(a_C). \end{align*} Since one factor in this product is zero, the whole product is zero. Thus every partition with a mixed-colour block disappears from the moment-cumulant expansion. After deleting exactly these zero terms, the moment is \begin{align*} \varphi(a_1\cdots a_n)=\sum_{\pi\in NC_c(n)}\prod_{B\in \pi}\kappa_{|B|}(a_B). \end{align*} This is the decisive reduction: freeness converts a potentially mixed cumulant expansion into a sum indexed only by colour-preserving noncrossing partitions. [/guided] [/step] [step:Identify each surviving cumulant as a marginal cumulant] Let $\pi\in NC_c(n)$ and let $B=\{b_1<\cdots<b_m\}$ be a block of $\pi$. Since $B$ has constant colour, there exists a unique index $i(B)\in I$ such that $i_{b_r}=i(B)$ for every $r\in\{1,\dots,m\}$. Hence \begin{align*} a_B=(a_{b_1},\dots,a_{b_m})\in \mathcal A_{i(B)}^m. \end{align*} Let $\varphi_{i(B)}:\mathcal A_{i(B)}\to \mathbb C$ be the restriction of $\varphi$ to $\mathcal A_{i(B)}$, and let $\kappa_m^{i(B)}:\mathcal A_{i(B)}^m\to \mathbb C$ be the $m$-th free cumulant computed in the marginal noncommutative probability space $(\mathcal A_{i(B)},\varphi_{i(B)})$. The free cumulant $\kappa_m(a_B)$ agrees with this marginal cumulant because the moment-cumulant recursion for the tuple $a_B$ uses only moments of products of entries from $\mathcal A_{i(B)}$, and those moments are identical whether computed by $\varphi$ on $\mathcal A$ or by its restriction $\varphi_{i(B)}$ on $\mathcal A_{i(B)}$. Therefore \begin{align*} \kappa_m(a_B)=\kappa_m^{i(B)}(a_B). \end{align*} Thus every factor in every surviving summand is a marginal free cumulant of one subalgebra. [/step] [step:Read the finite colour-preserving expansion as the required polynomial] Combining the preceding steps gives the explicit formula \begin{align*} \varphi(a_1\cdots a_n)=\sum_{\pi\in NC_c(n)}\prod_{B\in \pi}\kappa_{|B|}^{i(B)}(a_B). \end{align*} The right-hand side is a finite sum of finite products of marginal free cumulants. Therefore it is a polynomial with integer coefficients in those marginal cumulants. The indexing set $NC_c(n)$ depends only on the colour pattern $(i_1,\dots,i_n)$, so the polynomial form depends only on that pattern and not on any mixed cumulants. This proves that every mixed moment $\varphi(a_1\cdots a_n)$ is expressible as a polynomial in the marginal free cumulants of the free subalgebras. [/step]