[proofplan]
We start from the general [moment-cumulant formula for free cumulants](/theorems/7107), which expands $\varphi(a_1 \cdots a_n)$ as a sum over all noncrossing partitions of $\{1,\dots,n\}$. For each partition, the partition cumulant factors as a product of cumulants over its blocks. Speicher's criterion for freeness says that every mixed free cumulant involving variables from at least two different free subalgebras vanishes, so any partition with a non-constant-colour block contributes zero. The surviving terms are precisely the partitions all of whose blocks have constant colour.
[/proofplan]
[step:Expand the moment over all noncrossing partitions]
By the moment-cumulant formula for free cumulants, applied to the elements $a_1,\dots,a_n \in \mathcal A$, we have
\begin{align*}
\varphi(a_1 \cdots a_n)
= \sum_{\pi \in NC(n)} \kappa_\pi^\varphi(a_1,\dots,a_n).
\end{align*}
Here $NC(n)$ denotes the set of noncrossing partitions of $\{1,\dots,n\}$, and $\kappa_\pi^\varphi$ is the partition cumulant associated to the state $\varphi$.
[/step]
[step:Identify the partitions whose block cumulants vanish]
Let $\pi \in NC(n)$, and let $V = \{v_1 < \dots < v_m\}$ be a block of $\pi$. The contribution of this block to $\kappa_\pi^\varphi(a_1,\dots,a_n)$ is
\begin{align*}
\kappa_m^\varphi(a_{v_1},\dots,a_{v_m}).
\end{align*}
If $V$ is not constant-colour, then there exist $p,q \in V$ such that $i_p \neq i_q$. Thus the entries $a_{v_1},\dots,a_{v_m}$ are drawn from at least two distinct free subalgebras among the family $(\mathcal A_i)_{i \in I}$. By Speicher's criterion for freeness, this mixed free cumulant vanishes:
\begin{align*}
\kappa_m^\varphi(a_{v_1},\dots,a_{v_m}) = 0.
\end{align*}
Since $\kappa_\pi^\varphi(a_1,\dots,a_n)$ is the product of the block cumulants over all blocks of $\pi$, the vanishing of one block factor gives
\begin{align*}
\kappa_\pi^\varphi(a_1,\dots,a_n) = 0.
\end{align*}
[guided]
Fix a noncrossing partition $\pi \in NC(n)$. The partition cumulant $\kappa_\pi^\varphi(a_1,\dots,a_n)$ is not a new cumulant of order $n$; it is built block by block from the ordinary free cumulants. If the blocks of $\pi$ are $V_1,\dots,V_r$, and if a block $V_\ell$ is written in increasing order as
\begin{align*}
V_\ell = \{v_{\ell,1} < \dots < v_{\ell,m_\ell}\},
\end{align*}
then
\begin{align*}
\kappa_\pi^\varphi(a_1,\dots,a_n)
= \prod_{\ell=1}^r \kappa_{m_\ell}^\varphi(a_{v_{\ell,1}},\dots,a_{v_{\ell,m_\ell}}).
\end{align*}
Now suppose one block $V = \{v_1 < \dots < v_m\}$ is not constant-colour. This means that the colour map
\begin{align*}
c: \{1,\dots,n\} &\to I
\end{align*}
defined by $c(j) := i_j$ is not constant on $V$. Equivalently, there exist positions $p,q \in V$ such that $i_p \neq i_q$. Therefore the list of variables
\begin{align*}
a_{v_1},\dots,a_{v_m}
\end{align*}
contains entries from at least two distinct subalgebras in the free family $(\mathcal A_i)_{i \in I}$.
Speicher's criterion for freeness applies because the subalgebras $(\mathcal A_i)_{i \in I}$ are free with respect to $\varphi$ by hypothesis. The criterion says exactly that every free cumulant whose arguments come from more than one member of a free family is zero. Hence
\begin{align*}
\kappa_m^\varphi(a_{v_1},\dots,a_{v_m}) = 0.
\end{align*}
This block cumulant is one factor in the product defining $\kappa_\pi^\varphi(a_1,\dots,a_n)$. Since a product with a zero factor is zero, we obtain
\begin{align*}
\kappa_\pi^\varphi(a_1,\dots,a_n) = 0.
\end{align*}
Thus every noncrossing partition with at least one non-constant-colour block contributes nothing to the moment-cumulant expansion.
[/guided]
[/step]
[step:Restrict the moment-cumulant sum to constant-colour partitions]
The preceding step shows that every partition $\pi \in NC(n)$ with at least one non-constant-colour block has zero contribution. Therefore the full moment-cumulant sum reduces exactly to the subcollection of noncrossing partitions whose every block is constant-colour:
\begin{align*}
\varphi(a_1 \cdots a_n)
= \sum_{\substack{\pi \in NC(n), \text{every block of } \pi \text{ is constant-colour}}}
\kappa_\pi^\varphi(a_1,\dots,a_n).
\end{align*}
This is the desired formula.
[/step]
