[proofplan] We expand the moment using the standard free moment-cumulant formula for the cumulants associated to $\psi$. The semicircular cumulant hypothesis says that every cumulant of order different from two vanishes, so every noncrossing partition with a block of size other than two contributes zero. The only surviving noncrossing partitions are the noncrossing pair partitions, and each surviving pair block contributes the corresponding covariance entry. [/proofplan]

[step:Expand the word moment over noncrossing partitions]Let $NC(m)$ denote the set of noncrossing partitions of $\{1,\dots,m\}$. For a block $V=\{v_1<\cdots<v_r\}$ of a partition of $\{1,\dots,m\}$, define the block cumulant associated to the fixed word $s_{i_1}\cdots s_{i_m}$ by \begin{align*} \kappa_V:=\kappa_r(s_{i_{v_1}},\dots,s_{i_{v_r}}). \end{align*} The cumulants $\kappa_r$ are, by hypothesis, the free cumulants associated to $\psi$, and the entries $s_{i_1},\dots,s_{i_m}$ lie in $\mathcal A$. Therefore the standard free moment-cumulant formula applies to this word and gives \begin{align*} \psi(s_{i_1}\cdots s_{i_m})=\sum_{\pi\in NC(m)}\prod_{V\in\pi}\kappa_V. \end{align*}[/step]

[guided]We first name the ambient combinatorial set used in the moment-cumulant expansion. Let $NC(m)$ be the set of noncrossing partitions of $\{1,\dots,m\}$. If $V$ is a block of such a partition, write it in increasing order as $V=\{v_1<\cdots<v_r\}$. The order matters because the free cumulant $\kappa_r$ is an $r$-linear functional on ordered inputs. For the fixed word $s_{i_1}\cdots s_{i_m}$, define \begin{align*} \kappa_V:=\kappa_r(s_{i_{v_1}},\dots,s_{i_{v_r}}). \end{align*} The external input is the standard free moment-cumulant formula: for elements $a_1,\dots,a_m$ in a noncommutative probability space whose free cumulants are $\kappa_r$, one has \begin{align*} \psi(a_1\cdots a_m)=\sum_{\pi\in NC(m)}\prod_{V\in\pi}\kappa_{|V|}(a_j:j\in V \text{ in increasing order}). \end{align*} Its hypotheses are satisfied here because $(\mathcal A,\psi)$ is a noncommutative probability space, $s_{i_1},\dots,s_{i_m}\in\mathcal A$, and the maps $\kappa_r$ are stated to be the free cumulants associated to $\psi$. Applying the formula with $a_j=s_{i_j}$ for $1\leq j\leq m$ gives \begin{align*} \psi(s_{i_1}\cdots s_{i_m})=\sum_{\pi\in NC(m)}\prod_{V\in\pi}\kappa_V. \end{align*} This is the exact place where freeness-cumulant technology enters the proof: it converts the moment into a finite sum indexed by noncrossing partitions.[/guided]

[step:Use the semicircular cumulant hypothesis to keep only pair blocks] Fix $\pi\in NC(m)$. If $\pi$ has a block $V=\{v_1<\cdots<v_r\}$ with $r\neq 2$, then the semicircular-system hypothesis gives \begin{align*} \kappa_V=\kappa_r(s_{i_{v_1}},\dots,s_{i_{v_r}})=0. \end{align*} Hence the whole product $\prod_{V\in\pi}\kappa_V$ is zero unless every block of $\pi$ has size two. The partitions in $NC(m)$ whose blocks all have size two are exactly the elements of $NC_2(m)$. Therefore \begin{align*} \psi(s_{i_1}\cdots s_{i_m})=\sum_{\pi\in NC_2(m)}\prod_{\{p,q\}\in\pi}\kappa_2(s_{i_p},s_{i_q}), \end{align*} where each pair $\{p,q\}$ is read with $p<q$. [/step]

[step:Evaluate each surviving pair cumulant as a covariance entry] For every pair block $\{p,q\}$ with $p<q$, the covariance convention in the statement gives \begin{align*} \kappa_2(s_{i_p},s_{i_q})=c_{i_pi_q}. \end{align*} Substituting this identity into the restricted moment-cumulant sum yields \begin{align*} \psi(s_{i_1}\cdots s_{i_m})=\sum_{\pi\in NC_2(m)}\prod_{\{p,q\}\in\pi}c_{i_pi_q}. \end{align*} If $m$ is odd, no partition of $\{1,\dots,m\}$ into two-element blocks exists, so $NC_2(m)=\varnothing$ and the displayed sum is empty. With the usual convention that an empty sum is $0$, this gives $\psi(s_{i_1}\cdots s_{i_m})=0$ for odd $m$. This proves the stated Wick formula. [/step]