[proofplan] We prove the result by induction on the order $n$ of the cumulant. The case $n=2$ follows directly from the two-variable moment-cumulant formula and the normalization $\varphi(1_A)=1$. For the induction step, we delete the unit entry and compare the moment-cumulant expansion of $\varphi(x_1\cdots x_n)$ with the expansion of the shorter moment. In the expansion over noncrossing partitions, every partition except the one-block partition either has the unit as a singleton, in which case it matches a term in the shorter expansion, or has the unit inside a smaller nonsingleton block, in which case the induction hypothesis makes that block cumulant vanish. [/proofplan]

[step:Establish the two-variable case without using traciality] Let $x\in A$. Since $1_A$ is the multiplicative unit and $\varphi(1_A)=1$, the moment-cumulant formula for two variables gives $\varphi(x1_A)=\kappa_2(x,1_A)+\kappa_1(x)\kappa_1(1_A)$. The one-variable moment-cumulant formula gives $\kappa_1(y)=\varphi(y)$ for every $y\in A$, so $\kappa_1(1_A)=1$ and $\kappa_1(x)=\varphi(x)$. Since $x1_A=x$, the preceding identity becomes $\varphi(x)=\kappa_2(x,1_A)+\varphi(x)$. Subtracting $\varphi(x)$ gives $\kappa_2(x,1_A)=0$. Similarly, using $1_Ax=x$, the two-variable formula gives $\varphi(1_Ax)=\kappa_2(1_A,x)+\kappa_1(1_A)\kappa_1(x)$. Thus $\varphi(x)=\kappa_2(1_A,x)+\varphi(x)$, and hence $\kappa_2(1_A,x)=0$. This proves the assertion for $n=2$. [/step]

[step:Set up the induction and the deleted word] Fix an integer $n\geq 3$, and assume that the theorem has been proved for every cumulant order $m$ satisfying $2\leq m<n$. Let $x_1,\dots,x_n\in A$, and choose an index $j\in\{1,\dots,n\}$ such that $x_j=1_A$. Define the deleted word $y_1,\dots,y_{n-1}\in A$ by setting $y_r=x_r$ for $1\leq r<j$ and $y_r=x_{r+1}$ for $j\leq r\leq n-1$. Since $x_j=1_A$, multiplication in the unital algebra $A$ gives $x_1\cdots x_n=y_1\cdots y_{n-1}$. Applying $\varphi:A\to\mathbb{C}$ to both sides, we obtain $\varphi(x_1\cdots x_n)=\varphi(y_1\cdots y_{n-1})$. [/step]

[step:Separate the noncrossing partitions according to the block containing the unit]Let $NC(n)$ denote the set of noncrossing partitions of $\{1,\dots,n\}$. For a partition $\pi\in NC(n)$ and a block $V=\{i_1<\cdots<i_r\}$ of $\pi$, define $\kappa_V(x_1,\dots,x_n):=\kappa_r(x_{i_1},\dots,x_{i_r})$. Define the partition cumulant $\kappa_\pi(x_1,\dots,x_n):=\prod_{V\in\pi}\kappa_V(x_1,\dots,x_n)$. The product is a finite product of complex numbers, so its order is irrelevant. Let $1_n\in NC(n)$ be the one-block partition $\{\{1,\dots,n\}\}$. The moment-cumulant formula gives $\varphi(x_1\cdots x_n)=\sum_{\pi\in NC(n)}\kappa_\pi(x_1,\dots,x_n)$. We split this sum into three classes: the one-block partition $1_n$, partitions for which $\{j\}$ is a singleton block, and partitions different from $1_n$ for which the block containing $j$ has at least two elements.[/step]

[guided]The purpose of this step is to isolate exactly where the unknown cumulant $\kappa_n(x_1,\dots,x_n)$ appears. It appears only in the one-block partition $1_n$, because for that partition the unique block is $\{1,\dots,n\}$ and therefore $\kappa_{1_n}(x_1,\dots,x_n)=\kappa_n(x_1,\dots,x_n)$. We now define the notation used to discuss the other terms. The set $NC(n)$ is the set of noncrossing partitions of the ordered set $\{1,\dots,n\}$. If $\pi\in NC(n)$ and $V=\{i_1<\cdots<i_r\}$ is a block of $\pi$, then the block contributes the scalar $\kappa_V(x_1,\dots,x_n):=\kappa_r(x_{i_1},\dots,x_{i_r})$. The whole partition contributes the product over all its blocks: $\kappa_\pi(x_1,\dots,x_n):=\prod_{V\in\pi}\kappa_V(x_1,\dots,x_n)$. This product is taken in $\mathbb{C}$, since cumulants are complex-valued multilinear functionals, so the order of the blocks does not affect the value. The [moment-cumulant formula for free cumulants](/theorems/7107) says that $\varphi(x_1\cdots x_n)=\sum_{\pi\in NC(n)}\kappa_\pi(x_1,\dots,x_n)$. Because $x_j=1_A$, the decisive feature of a partition $\pi$ is the block containing $j$. There are three possibilities. First, $\pi=1_n$, which gives the target term $\kappa_n(x_1,\dots,x_n)$. Second, $\{j\}$ is a singleton block; those terms will reproduce the shorter moment expansion after deleting $x_j$. Third, $j$ lies in a block of size at least $2$ but $\pi\neq 1_n$; those terms contain a lower-order cumulant with a unit entry and will vanish by the induction hypothesis.[/guided]

[step:Show that nonsingleton proper blocks containing the unit give zero] Let $\pi\in NC(n)$ satisfy $\pi\neq 1_n$, and let $V\in\pi$ be the block containing $j$. Assume that $|V|\geq 2$. Since $\pi\neq 1_n$, the block $V$ is a proper subset of $\{1,\dots,n\}$, hence $2\leq |V|\leq n-1$. Write $V=\{i_1<\cdots<i_r\}$, where $r=|V|$. Since $j\in V$, there is an index $q\in\{1,\dots,r\}$ such that $i_q=j$. Therefore the block cumulant is $\kappa_V(x_1,\dots,x_n)=\kappa_r(x_{i_1},\dots,x_{i_{q-1}},1_A,x_{i_{q+1}},\dots,x_{i_r})$. The integer $r$ satisfies $2\leq r<n$, so the induction hypothesis applies to this $r$-th cumulant with a unit entry. Hence $\kappa_V(x_1,\dots,x_n)=0$. Since $\kappa_\pi(x_1,\dots,x_n)$ is a product containing this factor, we have $\kappa_\pi(x_1,\dots,x_n)=0$. [/step]

[step:Identify singleton-unit partitions with the shorter moment expansion] Let $NC_j^{\mathrm{sing}}(n)$ denote the set of partitions $\pi\in NC(n)$ such that $\{j\}$ is a block of $\pi$. Deleting the singleton block $\{j\}$ and then relabelling the ordered set $\{1,\dots,n\}\setminus\{j\}$ increasingly as $\{1,\dots,n-1\}$ defines a bijection $\rho:NC_j^{\mathrm{sing}}(n)\to NC(n-1)$. This map is bijective because inserting a singleton block at position $j$ into any noncrossing partition of $\{1,\dots,n-1\}$ preserves noncrossingness and is inverse to deletion. For every $\pi\in NC_j^{\mathrm{sing}}(n)$, the singleton block contributes $\kappa_{\{j\}}(x_1,\dots,x_n)=\kappa_1(1_A)=\varphi(1_A)=1$. All other blocks contribute exactly the corresponding block cumulants of $\rho(\pi)$ evaluated on $y_1,\dots,y_{n-1}$. Hence $\kappa_\pi(x_1,\dots,x_n)=\kappa_{\rho(\pi)}(y_1,\dots,y_{n-1})$. Summing over $NC_j^{\mathrm{sing}}(n)$ and using the bijection $\rho$, we obtain $\sum_{\pi\in NC_j^{\mathrm{sing}}(n)}\kappa_\pi(x_1,\dots,x_n)=\sum_{\sigma\in NC(n-1)}\kappa_\sigma(y_1,\dots,y_{n-1})$. By the moment-cumulant formula for the shorter word, $\sum_{\sigma\in NC(n-1)}\kappa_\sigma(y_1,\dots,y_{n-1})=\varphi(y_1\cdots y_{n-1})$. [/step]

[step:Compare the two expansions and isolate the one-block cumulant] Combining the partition decomposition, the vanishing of all nonsingleton proper-block terms containing $j$, and the singleton-block identification, we get $\varphi(x_1\cdots x_n)=\kappa_n(x_1,\dots,x_n)+\varphi(y_1\cdots y_{n-1})$. But from the deleted-word identity, $\varphi(x_1\cdots x_n)=\varphi(y_1\cdots y_{n-1})$. Subtracting the common term $\varphi(y_1\cdots y_{n-1})$ gives $\kappa_n(x_1,\dots,x_n)=0$. This completes the induction and proves the theorem for every $n\geq 2$. [/step]