[proofplan]
We expand the moment $\varphi(s_{i_1}\cdots s_{i_m})$ using the free moment-cumulant formula over noncrossing partitions. The defining cumulant condition for a semicircular family eliminates every partition that has a block of size different from $2$. Therefore no partition contributes in odd degree, while in even degree the surviving partitions are exactly the noncrossing pairings. For each surviving pairing, multiplicativity of the cumulant functional over blocks gives the product of the covariance entries.
[/proofplan]
[step:Identify the first and second cumulants of the semicircular family]
Fix $i,j \in I$. Since $(s_i)_{i \in I}$ is semicircular, its joint free cumulants vanish in every order other than $2$. Applying this with order $1$ gives
\begin{align*}
\kappa_1(s_i) = 0.
\end{align*}
By definition of the covariance matrix,
\begin{align*}
C_{ij} = \kappa_2(s_i,s_j).
\end{align*}
The free moment-cumulant formula applies because $\kappa_r$ denotes the joint free cumulants of the noncommutative probability space $(\mathcal{A},\varphi)$. In degree $2$, the noncrossing partitions of $\{1,2\}$ are $\{\{1,2\}\}$ and $\{\{1\},\{2\}\}$, so
\begin{align*}
\varphi(s_i s_j) = \kappa_2(s_i,s_j) + \kappa_1(s_i)\kappa_1(s_j).
\end{align*}
The first cumulants vanish, hence
\begin{align*}
\varphi(s_i s_j) = \kappa_2(s_i,s_j) = C_{ij}.
\end{align*}
[/step]
[step:Expand an arbitrary moment over noncrossing partitions]
Fix $m \in \mathbb{N}$ and indices $i_1,\dots,i_m \in I$. Let $NC(m)$ denote the set of noncrossing partitions of $\{1,\dots,m\}$. For a partition $\pi \in NC(m)$ and a block $V = \{v_1 < \cdots < v_r\}$ of $\pi$, define the block cumulant
\begin{align*}
\kappa_V(s_{i_1},\dots,s_{i_m}) := \kappa_r(s_{i_{v_1}},\dots,s_{i_{v_r}}).
\end{align*}
Define the partition cumulant
\begin{align*}
\kappa_\pi(s_{i_1},\dots,s_{i_m}) := \prod_{V \in \pi} \kappa_V(s_{i_1},\dots,s_{i_m}).
\end{align*}
The free moment-cumulant formula applies to the word $s_{i_1}\cdots s_{i_m}$ in the noncommutative probability space $(\mathcal{A},\varphi)$, with $\kappa_r$ the associated joint free cumulant maps. It gives
\begin{align*}
\varphi(s_{i_1}\cdots s_{i_m}) = \sum_{\pi \in NC(m)} \kappa_\pi(s_{i_1},\dots,s_{i_m}).
\end{align*}
[guided]
The purpose of this step is to translate the moment into cumulants, because the hypothesis defining a semicircular family is a statement about cumulants. We fix $m \in \mathbb{N}$ and indices $i_1,\dots,i_m \in I$, and we write $NC(m)$ for the set of noncrossing partitions of $\{1,\dots,m\}$.
For each noncrossing partition $\pi \in NC(m)$, the contribution of $\pi$ is built block by block. If $V = \{v_1 < \cdots < v_r\}$ is a block of $\pi$, define
\begin{align*}
\kappa_V(s_{i_1},\dots,s_{i_m}) := \kappa_r(s_{i_{v_1}},\dots,s_{i_{v_r}}).
\end{align*}
The order $v_1 < \cdots < v_r$ is the inherited order from $\{1,\dots,m\}$, so the variables are fed into the cumulant in the same order in which they occur in the word $s_{i_1}\cdots s_{i_m}$. The full contribution of $\pi$ is the product over all blocks:
\begin{align*}
\kappa_\pi(s_{i_1},\dots,s_{i_m}) := \prod_{V \in \pi} \kappa_V(s_{i_1},\dots,s_{i_m}).
\end{align*}
The free moment-cumulant formula states that the moment of a word is the sum of these partition cumulants over all noncrossing partitions. It applies here because $s_{i_1},\dots,s_{i_m}$ are elements of the noncommutative probability space $(\mathcal{A},\varphi)$ and $\kappa_r$ denotes the corresponding joint free cumulant maps. Applied to the word $s_{i_1}\cdots s_{i_m}$, it gives
\begin{align*}
\varphi(s_{i_1}\cdots s_{i_m}) = \sum_{\pi \in NC(m)} \kappa_\pi(s_{i_1},\dots,s_{i_m}).
\end{align*}
This is the exact point where noncrossing partitions enter the argument. The remaining work is to use the semicircular hypothesis to determine which terms in this sum can be nonzero.
[/guided]
[/step]
[step:Discard every partition with a block whose size is not two]
Let $\pi \in NC(m)$. Suppose $\pi$ has a block $V = \{v_1 < \cdots < v_r\}$ with $r \neq 2$. By the defining cumulant condition for a semicircular family,
\begin{align*}
\kappa_V(s_{i_1},\dots,s_{i_m}) = \kappa_r(s_{i_{v_1}},\dots,s_{i_{v_r}}) = 0.
\end{align*}
Since $\kappa_\pi(s_{i_1},\dots,s_{i_m})$ is the product of the block cumulants over all blocks of $\pi$, this zero factor implies
\begin{align*}
\kappa_\pi(s_{i_1},\dots,s_{i_m}) = 0.
\end{align*}
Thus a noncrossing partition can contribute to the moment sum only if every block has size $2$.
[guided]
The moment expansion is a sum over all noncrossing partitions, but the semicircular hypothesis says that only second cumulants can survive. Fix $\pi \in NC(m)$, and suppose that $\pi$ has a block $V = \{v_1 < \cdots < v_r\}$ with $r \neq 2$. By the definition of the block cumulant,
\begin{align*}
\kappa_V(s_{i_1},\dots,s_{i_m}) := \kappa_r(s_{i_{v_1}},\dots,s_{i_{v_r}}).
\end{align*}
Since $(s_i)_{i \in I}$ is semicircular, every joint free cumulant of order different from $2$ vanishes. Therefore
\begin{align*}
\kappa_V(s_{i_1},\dots,s_{i_m}) = 0.
\end{align*}
The partition cumulant $\kappa_\pi(s_{i_1},\dots,s_{i_m})$ is the product of all block cumulants attached to blocks of $\pi$. A product with a zero factor is zero, so
\begin{align*}
\kappa_\pi(s_{i_1},\dots,s_{i_m}) = 0.
\end{align*}
Thus the only noncrossing partitions that can contribute to the moment are those whose every block has size $2$.
[/guided]
[/step]
[step:Conclude that all odd moments vanish]
Assume $m$ is odd. A partition of $\{1,\dots,m\}$ whose every block has size $2$ cannot exist, because such a partition would write the odd integer $m$ as a sum of twos. Therefore every $\pi \in NC(m)$ has at least one block whose size is not $2$, and the previous step gives
\begin{align*}
\kappa_\pi(s_{i_1},\dots,s_{i_m}) = 0.
\end{align*}
Substituting into the moment-cumulant expansion yields
\begin{align*}
\varphi(s_{i_1}\cdots s_{i_m}) = \sum_{\pi \in NC(m)} 0 = 0.
\end{align*}
[guided]
Assume that $m$ is odd. If every block of a partition of $\{1,\dots,m\}$ had size $2$, then the total number of elements would be twice the number of blocks, hence even. This contradicts the oddness of $m$. Therefore every partition $\pi \in NC(m)$ has at least one block whose size is not $2$.
By the previous elimination argument, each such partition satisfies
\begin{align*}
\kappa_\pi(s_{i_1},\dots,s_{i_m}) = 0.
\end{align*}
Substituting these zero contributions into the free moment-cumulant expansion gives
\begin{align*}
\varphi(s_{i_1}\cdots s_{i_m}) = \sum_{\pi \in NC(m)} 0 = 0.
\end{align*}
This proves the vanishing of all odd moments.
[/guided]
[/step]
[step:Sum the surviving pairings in even degree]
Assume $m = 2n$ for some $n \in \mathbb{N}$. By the preceding elimination step, the only nonzero terms in the moment-cumulant expansion come from those $\pi \in NC(2n)$ all of whose blocks have size $2$. This set is precisely $NC_2(2n)$.
Fix $\pi \in NC_2(2n)$. For every block $\{p,q\} \in \pi$, write the elements in increasing order when evaluating the block cumulant. Since $\kappa_2$ is the second joint cumulant and $C_{ab} = \kappa_2(s_a,s_b)$, the block contribution is
\begin{align*}
\kappa_{\{p,q\}}(s_{i_1},\dots,s_{i_{2n}}) = C_{i_p i_q}.
\end{align*}
Therefore multiplicativity over blocks gives
\begin{align*}
\kappa_\pi(s_{i_1},\dots,s_{i_{2n}}) = \prod_{\{p,q\} \in \pi} C_{i_p i_q}.
\end{align*}
Substituting the surviving terms into the free moment-cumulant expansion gives
\begin{align*}
\varphi(s_{i_1}\cdots s_{i_{2n}}) = \sum_{\pi \in NC_2(2n)} \prod_{\{p,q\} \in \pi} C_{i_p i_q}.
\end{align*}
This is the desired free Wick formula.
[guided]
Now assume $m = 2n$ for some $n \in \mathbb{N}$. The elimination step says that a noncrossing partition contributes to the moment expansion only when all of its blocks have size $2$. Among the noncrossing partitions of $\{1,\dots,2n\}$, this surviving set is exactly $NC_2(2n)$, the set of noncrossing pair partitions.
Fix $\pi \in NC_2(2n)$. Each block of $\pi$ has the form $\{p,q\}$ with $p \neq q$. When evaluating its block cumulant, we use the inherited order from the word $s_{i_1}\cdots s_{i_{2n}}$, so with $p < q$ the block contribution is
\begin{align*}
\kappa_{\{p,q\}}(s_{i_1},\dots,s_{i_{2n}}) = \kappa_2(s_{i_p},s_{i_q}).
\end{align*}
By the definition of the covariance entries, $C_{ab} := \kappa_2(s_a,s_b)$ for all $a,b \in I$. Hence
\begin{align*}
\kappa_{\{p,q\}}(s_{i_1},\dots,s_{i_{2n}}) = C_{i_p i_q}.
\end{align*}
Because the partition cumulant is the product of the block cumulants, we obtain
\begin{align*}
\kappa_\pi(s_{i_1},\dots,s_{i_{2n}}) = \prod_{\{p,q\} \in \pi} C_{i_p i_q}.
\end{align*}
Finally, substituting exactly these surviving terms into the free moment-cumulant expansion gives
\begin{align*}
\varphi(s_{i_1}\cdots s_{i_{2n}}) = \sum_{\pi \in NC_2(2n)} \prod_{\{p,q\} \in \pi} C_{i_p i_q}.
\end{align*}
This proves the even-degree formula and completes the proof.
[/guided]
[/step]
