[proofplan] We construct the cumulants recursively by separating, in the desired moment-cumulant identity, the contribution of the one-block partition from all other noncrossing partitions. For a partition of $\{1,\dots,n\}$ different from the one-block partition, every block has size strictly smaller than $n$, so its contribution is already defined at the $n$th stage of the recursion. This gives existence and multilinearity. Uniqueness follows from the same separation by induction: after all lower-order cumulants are fixed, the one-block term is the only remaining unknown. [/proofplan]

[step:Introduce the one-block partition and the moment maps] For each integer $n\geq 1$, let $\hat{1}_n\in NC(n)$ denote the partition with the single block $\{1,\dots,n\}$. Define the $n$th moment map $M_n:\mathcal A^n\to\mathbb C$ by \begin{align*} M_n(a_1,\dots,a_n):=\varphi(a_1\cdots a_n). \end{align*} The multiplication map $m_n:\mathcal A^n\to\mathcal A$, $(a_1,\dots,a_n)\mapsto a_1\cdots a_n$, is multilinear by repeated bilinearity of multiplication in the complex algebra $\mathcal A$. Since $\varphi:\mathcal A\to\mathbb C$ is linear, $M_n=\varphi\circ m_n$ is multilinear. [/step]

[step:Construct the cumulants recursively]We define $\kappa_n:\mathcal A^n\to\mathbb C$ by induction on $n$. For $n=1$, the lattice $NC(1)$ has only $\hat{1}_1$, so define \begin{align*} \kappa_1(a_1):=M_1(a_1)=\varphi(a_1). \end{align*} This map is linear because $M_1$ is linear. Assume now that $n\geq 2$ and that multilinear maps $\kappa_r:\mathcal A^r\to\mathbb C$ have already been defined for every integer $1\leq r<n$. For $\pi\in NC(n)$ with $\pi\neq\hat{1}_n$, each block $V\in\pi$ has cardinality $|V|<n$; hence $\kappa_V[a_1,\dots,a_n]$ is already defined. Define the lower-order contribution $R_n:\mathcal A^n\to\mathbb C$ by \begin{align*} R_n(a_1,\dots,a_n):=\sum_{\pi\in NC(n),\,\pi\neq\hat{1}_n}\prod_{V\in\pi}\kappa_V[a_1,\dots,a_n]. \end{align*} Then define \begin{align*} \kappa_n(a_1,\dots,a_n):=M_n(a_1,\dots,a_n)-R_n(a_1,\dots,a_n). \end{align*} For each fixed $\pi\neq\hat{1}_n$, every variable $a_j$ occurs in exactly one block factor, and that block factor is multilinear in the variables indexed by the block. All other block factors are scalar-valued functions of the remaining variables, so the finite product is multilinear in $(a_1,\dots,a_n)$. Since $NC(n)$ is finite, $R_n$ is a finite sum of multilinear maps and is therefore multilinear. Thus $\kappa_n=M_n-R_n$ is multilinear.[/step]

[guided]The recursion is designed to make the desired formula true at level $n$ by definition. The only possible obstruction is whether the right-hand side can already be evaluated before $\kappa_n$ has been defined. This is where the one-block partition is separated out. Let $\hat{1}_n\in NC(n)$ be the partition whose only block is $\{1,\dots,n\}$. If $\pi\in NC(n)$ and $\pi\neq\hat{1}_n$, then no block of $\pi$ can have cardinality $n$, because a block of cardinality $n$ would be the whole set $\{1,\dots,n\}$ and hence would force $\pi=\hat{1}_n$. Therefore every block $V\in\pi$ has cardinality $|V|<n$. By the induction hypothesis, the cumulant map $\kappa_{|V|}:\mathcal A^{|V|}\to\mathbb C$ is already available. For $n=1$, there is no lower-order contribution. We set \begin{align*} \kappa_1(a_1):=\varphi(a_1). \end{align*} This is the only possible definition because the moment-cumulant formula for $n=1$ reads $\varphi(a_1)=\kappa_1(a_1)$. Now fix $n\geq 2$ and assume that $\kappa_r$ has been defined as a multilinear map $\mathcal A^r\to\mathbb C$ for every $1\leq r<n$. Define the contribution of all non-one-block partitions by \begin{align*} R_n(a_1,\dots,a_n):=\sum_{\pi\in NC(n),\,\pi\neq\hat{1}_n}\prod_{V\in\pi}\kappa_V[a_1,\dots,a_n]. \end{align*} This expression is well-defined because each factor uses only cumulants of order strictly smaller than $n$. The sum is finite because $NC(n)$ is finite by hypothesis. We next check multilinearity, since the theorem asserts that the cumulants are multilinear maps. The moment map $M_n:\mathcal A^n\to\mathbb C$ given by $M_n(a_1,\dots,a_n)=\varphi(a_1\cdots a_n)$ is multilinear: the product map $(a_1,\dots,a_n)\mapsto a_1\cdots a_n$ is multilinear by bilinearity of multiplication in $\mathcal A$, and applying the linear map $\varphi$ preserves multilinearity. For a fixed partition $\pi\neq\hat{1}_n$, each variable $a_j$ appears in exactly one block $V$ of $\pi$. The factor attached to that block is multilinear in its own variables, and the remaining block factors are scalar-valued functions of the other variables. Hence the product over blocks is linear in each variable separately. Therefore every summand defining $R_n$ is multilinear, and the finite sum $R_n$ is multilinear. We now define the missing one-block contribution by subtracting all lower-order contributions from the moment: \begin{align*} \kappa_n(a_1,\dots,a_n):=M_n(a_1,\dots,a_n)-R_n(a_1,\dots,a_n). \end{align*} Since both $M_n$ and $R_n$ are multilinear, this definition makes $\kappa_n:\mathcal A^n\to\mathbb C$ multilinear. The point of the definition is that the only term not included in $R_n$ is the one-block term, and the one-block term is exactly $\kappa_n(a_1,\dots,a_n)$.[/guided]

[step:Verify the moment-cumulant identity for the constructed maps] We prove by induction on $n$ that the constructed maps satisfy the displayed formula. For $n=1$, the identity is \begin{align*} \varphi(a_1)=\kappa_1(a_1), \end{align*} which holds by the definition of $\kappa_1$. Let $n\geq 2$ and assume the maps have been constructed up to order $n$. By the definition of $R_n$ and the definition of the one-block contribution, \begin{align*} \sum_{\pi\in NC(n)}\kappa_\pi[a_1,\dots,a_n]=R_n(a_1,\dots,a_n)+\kappa_n(a_1,\dots,a_n). \end{align*} Using the recursive definition of $\kappa_n$, the right-hand side is \begin{align*} R_n(a_1,\dots,a_n)+M_n(a_1,\dots,a_n)-R_n(a_1,\dots,a_n)=M_n(a_1,\dots,a_n). \end{align*} Since $M_n(a_1,\dots,a_n)=\varphi(a_1\cdots a_n)$, the desired formula holds for this $n$. Thus the formula holds for every $n\geq 1$. [/step]

[step:Prove uniqueness by isolating the one-block term] Let $(\lambda_n)_{n\geq 1}$ be another family of multilinear maps $\lambda_n:\mathcal A^n\to\mathbb C$ satisfying the same moment-cumulant formula, with the analogous block notation $\lambda_\pi[a_1,\dots,a_n]$. We prove $\lambda_n=\kappa_n$ for all $n\geq 1$ by induction on $n$. For $n=1$, the formula gives \begin{align*} \lambda_1(a_1)=\varphi(a_1)=\kappa_1(a_1), \end{align*} so $\lambda_1=\kappa_1$. Assume that $n\geq 2$ and that $\lambda_r=\kappa_r$ for every $1\leq r<n$. For arbitrary $a_1,\dots,a_n\in\mathcal A$, subtract the two moment-cumulant formulas at level $n$: \begin{align*} 0=\sum_{\pi\in NC(n)}\lambda_\pi[a_1,\dots,a_n]-\sum_{\pi\in NC(n)}\kappa_\pi[a_1,\dots,a_n]. \end{align*} If $\pi\neq\hat{1}_n$, every block of $\pi$ has size smaller than $n$, so the induction hypothesis gives $\lambda_\pi[a_1,\dots,a_n]=\kappa_\pi[a_1,\dots,a_n]$. All non-one-block terms cancel, leaving \begin{align*} 0=\lambda_n(a_1,\dots,a_n)-\kappa_n(a_1,\dots,a_n). \end{align*} Since the elements $a_1,\dots,a_n\in\mathcal A$ were arbitrary, $\lambda_n=\kappa_n$. Induction proves uniqueness of the whole family $(\kappa_n)_{n\geq 1}$. [/step]

[step:Conclude existence and uniqueness of the free cumulants] The recursive construction gives a multilinear map $\kappa_n:\mathcal A^n\to\mathbb C$ for every $n\geq 1$, and the preceding verification proves that these maps satisfy the Speicher moment-cumulant formula over $NC(n)$ for every $n$. The induction argument shows that no other family of multilinear maps can satisfy the same formula. Therefore the free cumulants of $(\mathcal A,\varphi)$ exist and are unique. [/step]