[proofplan]
We expand the scalar moment $\varphi(x_1x_2y_1y_2)$ by the moment-cumulant formula over $NC(4)$. We also expand $\varphi(x_1x_2)\varphi(y_1y_2)$, and identify this product with the contribution of exactly those noncrossing partitions whose blocks stay inside the two intervals $\{1,2\}$ and $\{3,4\}$. Subtracting the two expansions leaves precisely the partitions that connect the two intervals, which is equivalent to the join condition $\pi\vee\rho=1_4$.
[/proofplan]
[step:Expand the fourfold moment over all noncrossing partitions]
Let $a_1,a_2,a_3,a_4\in\mathcal A$ be defined by $a_1=x_1$, $a_2=x_2$, $a_3=y_1$, and $a_4=y_2$. By the [moment-cumulant formula for free cumulants](/theorems/7107) (citing a result not yet in the wiki: Moment-cumulant formula), applied to the ordered tuple $(a_1,a_2,a_3,a_4)$, we have
\begin{align*}
\varphi(x_1x_2y_1y_2)=\sum_{\pi\in NC(4)}\kappa_\pi(x_1,x_2,y_1,y_2).
\end{align*}
[/step]
[step:Identify the product of the two shorter moments with interval-contained partitions]
Apply the moment-cumulant formula to the two ordered pairs $(x_1,x_2)$ and $(y_1,y_2)$. This gives
\begin{align*}
\varphi(x_1x_2)=\sum_{\sigma\in NC(2)}\kappa_\sigma(x_1,x_2).
\end{align*}
and
\begin{align*}
\varphi(y_1y_2)=\sum_{\tau\in NC(2)}\kappa_\tau(y_1,y_2).
\end{align*}
Multiplying the two finite sums yields
\begin{align*}
\varphi(x_1x_2)\varphi(y_1y_2)=\sum_{\sigma\in NC(2)}\sum_{\tau\in NC(2)}\kappa_\sigma(x_1,x_2)\kappa_\tau(y_1,y_2).
\end{align*}
Each pair $(\sigma,\tau)\in NC(2)\times NC(2)$ determines a partition $\pi\in NC(4)$ by placing the blocks of $\sigma$ on $\{1,2\}$ and the blocks of $\tau$ on $\{3,4\}$. Equivalently, these are exactly the partitions $\pi\in NC(4)$ such that every block of $\pi$ is contained in one of the two blocks of $\rho$. For such $\pi$, multiplicativity of $\kappa_\pi$ over blocks gives
\begin{align*}
\kappa_\pi(x_1,x_2,y_1,y_2)=\kappa_\sigma(x_1,x_2)\kappa_\tau(y_1,y_2).
\end{align*}
Therefore
\begin{align*}
\varphi(x_1x_2)\varphi(y_1y_2)=\sum_{\substack{\pi\in NC(4), \text{every block of }\pi\text{ is contained in }\{1, 2\}\text{ or }\{3, 4\}}}\kappa_\pi(x_1,x_2,y_1,y_2).
\end{align*}
[guided]
The product $\varphi(x_1x_2)\varphi(y_1y_2)$ should be viewed as a cumulant expansion in which the first two positions and the last two positions are not allowed to interact. Applying the moment-cumulant formula to the first pair gives
\begin{align*}
\varphi(x_1x_2)=\sum_{\sigma\in NC(2)}\kappa_\sigma(x_1,x_2).
\end{align*}
Applying the same formula to the second pair gives
\begin{align*}
\varphi(y_1y_2)=\sum_{\tau\in NC(2)}\kappa_\tau(y_1,y_2).
\end{align*}
Multiplying these two expansions gives
\begin{align*}
\varphi(x_1x_2)\varphi(y_1y_2)=\sum_{\sigma\in NC(2)}\sum_{\tau\in NC(2)}\kappa_\sigma(x_1,x_2)\kappa_\tau(y_1,y_2).
\end{align*}
Now we translate this double sum into a single sum over $NC(4)$. Given $\sigma\in NC(2)$ and $\tau\in NC(2)$, form a partition $\pi$ of $\{1,2,3,4\}$ by using the blocks of $\sigma$ on the first interval $\{1,2\}$ and the blocks of $\tau$ on the second interval $\{3,4\}$. This partition is noncrossing because it is the disjoint union of two noncrossing partitions on adjacent intervals. Its blocks never mix an index from $\{1,2\}$ with an index from $\{3,4\}$.
Conversely, if $\pi\in NC(4)$ has every block contained in one of the two intervals $\{1,2\}$ and $\{3,4\}$, then restricting $\pi$ to $\{1,2\}$ gives a partition $\sigma\in NC(2)$ and restricting $\pi$ to $\{3,4\}$ gives a partition $\tau\in NC(2)$. Thus the double sum over $(\sigma,\tau)$ is exactly the sum over interval-contained partitions $\pi\in NC(4)$. For such $\pi$, the definition of the multiplicative cumulant over blocks gives
\begin{align*}
\kappa_\pi(x_1,x_2,y_1,y_2)=\kappa_\sigma(x_1,x_2)\kappa_\tau(y_1,y_2).
\end{align*}
Hence
\begin{align*}
\varphi(x_1x_2)\varphi(y_1y_2)=\sum_{\substack{\pi\in NC(4), \text{every block of }\pi\text{ is contained in }\{1, 2\}\text{ or }\{3, 4\}}}\kappa_\pi(x_1,x_2,y_1,y_2).
\end{align*}
[/guided]
[/step]
[step:Subtract the separated contribution from the fourfold moment]
By the defining relation between the second free cumulant and moments of two variables, applied to the two elements $x_1x_2\in\mathcal A$ and $y_1y_2\in\mathcal A$, we have
\begin{align*}
\kappa_2(x_1x_2,y_1y_2)=\varphi(x_1x_2y_1y_2)-\varphi(x_1x_2)\varphi(y_1y_2).
\end{align*}
Substituting the two expansions obtained above gives
\begin{align*}
\kappa_2(x_1x_2,y_1y_2)=\sum_{\pi\in NC(4)}\kappa_\pi(x_1,x_2,y_1,y_2)-\sum_{\substack{\pi\in NC(4), \text{every block of }\pi\text{ is contained in }\{1, 2\}\text{ or }\{3, 4\}}}\kappa_\pi(x_1,x_2,y_1,y_2).
\end{align*}
Therefore the remaining terms are exactly those $\pi\in NC(4)$ having at least one block that meets both intervals $\{1,2\}$ and $\{3,4\}$.
[/step]
[step:Translate interval connectivity into the join condition]
We claim that, for $\pi\in NC(4)$, the partition $\pi$ has at least one block meeting both $\{1,2\}$ and $\{3,4\}$ if and only if $\pi\vee\rho=1_4$.
If every block of $\pi$ is contained in one of the two blocks of $\rho$, then $\pi$ refines $\rho$, so the least common coarsening of $\pi$ and $\rho$ is $\rho$, not $1_4$. Thus $\pi\vee\rho\ne 1_4$.
Conversely, suppose that some block $V$ of $\pi$ meets both $\{1,2\}$ and $\{3,4\}$. Since $\rho$ already identifies $1$ with $2$ and identifies $3$ with $4$, the additional block $V$ connects the two blocks of $\rho$ into a single equivalence class. Hence the least common coarsening of $\pi$ and $\rho$ is the one-block partition $1_4$.
Combining this equivalence with the previous step gives
\begin{align*}
\kappa_2(x_1x_2,y_1y_2)=\sum_{\substack{\pi\in NC(4), \pi\vee\rho=1_4}}\kappa_\pi(x_1,x_2,y_1,y_2).
\end{align*}
This is the desired identity.
[/step]
