[proofplan] We use the Speicher moment-cumulant formula in its defining form for the free cumulants associated to $\varphi$. For each noncrossing partition, we compute the product of cumulants over its blocks, with the variables in each block taken in increasing order. The proof is then a finite enumeration of $NC(1)$, $NC(2)$, $NC(3)$, and $NC(4)$; in order four the only pair-pair partition excluded is the crossing partition $\{\{1,3\},\{2,4\}\}$. [/proofplan]

[step:Fix the partition cumulant convention] For $n\in\{1,2,3,4\}$, let $NC(n)$ denote the set of noncrossing partitions of $\{1,\ldots,n\}$. For $\pi\in NC(n)$, define $\kappa_\pi(a_1,\ldots,a_n)$ as follows. If $V=\{i_1<\cdots<i_r\}$ is a block of $\pi$, its block contribution is $\kappa_r(a_{i_1},\ldots,a_{i_r})$, and $\kappa_\pi(a_1,\ldots,a_n)$ is the product in $\mathbb C$ of these block contributions over all blocks $V$ of $\pi$. Since the theorem assumes that $\kappa_n$ are the free cumulants associated to $\varphi$ through the Speicher moment-cumulant formula, for every $n\in\{1,2,3,4\}$ we have \begin{align*} \varphi(a_1\cdots a_n)=\sum_{\pi\in NC(n)}\kappa_\pi(a_1,\ldots,a_n). \end{align*} This is the exact external input used below. [/step]

[step:Enumerate the noncrossing partitions and substitute their block cumulants]For $n=1$, the only noncrossing partition is $\{\{1\}\}$. The Speicher formula gives \begin{align*} \varphi(a_1)=\kappa_1(a_1). \end{align*} For $n=2$, the noncrossing partitions are $\{\{1,2\}\}$ and $\{\{1\},\{2\}\}$. Therefore \begin{align*} \varphi(a_1a_2)=\kappa_2(a_1,a_2)+\kappa_1(a_1)\kappa_1(a_2). \end{align*} For $n=3$, every partition of $\{1,2,3\}$ is noncrossing. These five partitions are $\{\{1,2,3\}\}$, $\{\{1,2\},\{3\}\}$, $\{\{1,3\},\{2\}\}$, $\{\{2,3\},\{1\}\}$, and $\{\{1\},\{2\},\{3\}\}$. Substitution gives \begin{align*} \varphi(a_1a_2a_3)=\kappa_3(a_1,a_2,a_3)+\kappa_2(a_1,a_2)\kappa_1(a_3)+\kappa_2(a_1,a_3)\kappa_1(a_2)+\kappa_2(a_2,a_3)\kappa_1(a_1)+\kappa_1(a_1)\kappa_1(a_2)\kappa_1(a_3). \end{align*} For $n=4$, the noncrossing partitions of $\{1,2,3,4\}$ are the following fourteen partitions: $\{\{1,2,3,4\}\}$; the four partitions $\{\{1,2,3\},\{4\}\}$, $\{\{1,2,4\},\{3\}\}$, $\{\{1,3,4\},\{2\}\}$, and $\{\{2,3,4\},\{1\}\}$; the two noncrossing pair-pair partitions $\{\{1,2\},\{3,4\}\}$ and $\{\{1,4\},\{2,3\}\}$; the six pair-singleton-singleton partitions $\{\{1,2\},\{3\},\{4\}\}$, $\{\{1,3\},\{2\},\{4\}\}$, $\{\{1,4\},\{2\},\{3\}\}$, $\{\{2,3\},\{1\},\{4\}\}$, $\{\{2,4\},\{1\},\{3\}\}$, and $\{\{3,4\},\{1\},\{2\}\}$; and the all-singleton partition $\{\{1\},\{2\},\{3\},\{4\}\}$. The pair-pair partition $\{\{1,3\},\{2,4\}\}$ is not included because it is crossing. Applying the Speicher formula to this list gives exactly \begin{align*} \varphi(a_1a_2a_3a_4)=\kappa_4(a_1,a_2,a_3,a_4)+\kappa_3(a_1,a_2,a_3)\kappa_1(a_4)+\kappa_3(a_1,a_2,a_4)\kappa_1(a_3)+\kappa_3(a_1,a_3,a_4)\kappa_1(a_2)+\kappa_3(a_2,a_3,a_4)\kappa_1(a_1)+\kappa_2(a_1,a_2)\kappa_2(a_3,a_4)+\kappa_2(a_1,a_4)\kappa_2(a_2,a_3)+\kappa_2(a_1,a_2)\kappa_1(a_3)\kappa_1(a_4)+\kappa_2(a_1,a_3)\kappa_1(a_2)\kappa_1(a_4)+\kappa_2(a_1,a_4)\kappa_1(a_2)\kappa_1(a_3)+\kappa_2(a_2,a_3)\kappa_1(a_1)\kappa_1(a_4)+\kappa_2(a_2,a_4)\kappa_1(a_1)\kappa_1(a_3)+\kappa_2(a_3,a_4)\kappa_1(a_1)\kappa_1(a_2)+\kappa_1(a_1)\kappa_1(a_2)\kappa_1(a_3)\kappa_1(a_4). \end{align*} These are the four asserted identities.[/step]

[guided]The only theorem being used is the Speicher moment-cumulant formula, and the theorem statement supplies exactly the hypotheses needed for it: $(\mathcal A,\varphi)$ is a noncommutative probability space and $\kappa_n$ are the associated free cumulants. For a partition $\pi\in NC(n)$, the formula says \begin{align*} \varphi(a_1\cdots a_n)=\sum_{\pi\in NC(n)}\kappa_\pi(a_1,\ldots,a_n), \end{align*} where each block $V=\{i_1<\cdots<i_r\}$ contributes $\kappa_r(a_{i_1},\ldots,a_{i_r})$. Thus the proof reduces to listing the relevant noncrossing partitions and translating each block into its cumulant factor. For $n=1$, there is only one partition, namely $\{\{1\}\}$. Its unique block contributes $\kappa_1(a_1)$, so the formula gives \begin{align*} \varphi(a_1)=\kappa_1(a_1). \end{align*} For $n=2$, the two noncrossing partitions are $\{\{1,2\}\}$ and $\{\{1\},\{2\}\}$. The first has one block and contributes $\kappa_2(a_1,a_2)$; the second has two singleton blocks and contributes $\kappa_1(a_1)\kappa_1(a_2)$. Hence \begin{align*} \varphi(a_1a_2)=\kappa_2(a_1,a_2)+\kappa_1(a_1)\kappa_1(a_2). \end{align*} For $n=3$, no crossing can occur among partitions of three ordered points, so all five partitions of $\{1,2,3\}$ belong to $NC(3)$. Their block contributions are as follows: $\{\{1,2,3\}\}$ contributes $\kappa_3(a_1,a_2,a_3)$; $\{\{1,2\},\{3\}\}$ contributes $\kappa_2(a_1,a_2)\kappa_1(a_3)$; $\{\{1,3\},\{2\}\}$ contributes $\kappa_2(a_1,a_3)\kappa_1(a_2)$; $\{\{2,3\},\{1\}\}$ contributes $\kappa_2(a_2,a_3)\kappa_1(a_1)$; and $\{\{1\},\{2\},\{3\}\}$ contributes $\kappa_1(a_1)\kappa_1(a_2)\kappa_1(a_3)$. Summing these contributions gives \begin{align*} \varphi(a_1a_2a_3)=\kappa_3(a_1,a_2,a_3)+\kappa_2(a_1,a_2)\kappa_1(a_3)+\kappa_2(a_1,a_3)\kappa_1(a_2)+\kappa_2(a_2,a_3)\kappa_1(a_1)+\kappa_1(a_1)\kappa_1(a_2)\kappa_1(a_3). \end{align*} For $n=4$, we separate the noncrossing partitions by block sizes. The one-block partition $\{\{1,2,3,4\}\}$ contributes $\kappa_4(a_1,a_2,a_3,a_4)$. The four partitions with one three-element block and one singleton contribute $\kappa_3(a_1,a_2,a_3)\kappa_1(a_4)$, $\kappa_3(a_1,a_2,a_4)\kappa_1(a_3)$, $\kappa_3(a_1,a_3,a_4)\kappa_1(a_2)$, and $\kappa_3(a_2,a_3,a_4)\kappa_1(a_1)$. For pair-pair partitions, $\{\{1,2\},\{3,4\}\}$ and $\{\{1,4\},\{2,3\}\}$ are noncrossing, while $\{\{1,3\},\{2,4\}\}$ is crossing because the indices interlace as $1<2<3<4$ with $1,3$ in one block and $2,4$ in another. Thus the pair-pair contributions are $\kappa_2(a_1,a_2)\kappa_2(a_3,a_4)$ and $\kappa_2(a_1,a_4)\kappa_2(a_2,a_3)$. The six partitions with one pair and two singletons contribute $\kappa_2(a_1,a_2)\kappa_1(a_3)\kappa_1(a_4)$, $\kappa_2(a_1,a_3)\kappa_1(a_2)\kappa_1(a_4)$, $\kappa_2(a_1,a_4)\kappa_1(a_2)\kappa_1(a_3)$, $\kappa_2(a_2,a_3)\kappa_1(a_1)\kappa_1(a_4)$, $\kappa_2(a_2,a_4)\kappa_1(a_1)\kappa_1(a_3)$, and $\kappa_2(a_3,a_4)\kappa_1(a_1)\kappa_1(a_2)$. Finally, the all-singleton partition contributes $\kappa_1(a_1)\kappa_1(a_2)\kappa_1(a_3)\kappa_1(a_4)$. Summing all fourteen contributions gives \begin{align*} \varphi(a_1a_2a_3a_4)=\kappa_4(a_1,a_2,a_3,a_4)+\kappa_3(a_1,a_2,a_3)\kappa_1(a_4)+\kappa_3(a_1,a_2,a_4)\kappa_1(a_3)+\kappa_3(a_1,a_3,a_4)\kappa_1(a_2)+\kappa_3(a_2,a_3,a_4)\kappa_1(a_1)+\kappa_2(a_1,a_2)\kappa_2(a_3,a_4)+\kappa_2(a_1,a_4)\kappa_2(a_2,a_3)+\kappa_2(a_1,a_2)\kappa_1(a_3)\kappa_1(a_4)+\kappa_2(a_1,a_3)\kappa_1(a_2)\kappa_1(a_4)+\kappa_2(a_1,a_4)\kappa_1(a_2)\kappa_1(a_3)+\kappa_2(a_2,a_3)\kappa_1(a_1)\kappa_1(a_4)+\kappa_2(a_2,a_4)\kappa_1(a_1)\kappa_1(a_3)+\kappa_2(a_3,a_4)\kappa_1(a_1)\kappa_1(a_2)+\kappa_1(a_1)\kappa_1(a_2)\kappa_1(a_3)\kappa_1(a_4). \end{align*} This proves the fourth identity and completes the derivation of all four displayed formulas.[/guided]