[proofplan]
We prove invariance under one cyclic rotation and then iterate. If $w=UV$ and $v=VU$, evaluation at $a_1,\dots,a_d$ turns these words into the products $U(a_1,\dots,a_d)V(a_1,\dots,a_d)$ and $V(a_1,\dots,a_d)U(a_1,\dots,a_d)$ in $\mathcal A$. The tracial identity $\varphi(xy)=\varphi(yx)$ then gives equality of the corresponding moments. Since cyclic equivalence is generated by finitely many such rotations, the equality propagates along the finite chain from $w$ to $v$.
[/proofplan]
[step:Reduce cyclic equivalence to a finite chain of single rotations]
Let $\mathcal W_d$ denote the set of all words in the noncommuting indeterminates $X_1,\dots,X_d$, including the empty word. By the definition of cyclic equivalence, there exist an integer $m \geq 0$ and words $w_0,w_1,\dots,w_m \in \mathcal W_d$ with $w_0=w$, $w_m=v$, such that for each $r \in \{0,\dots,m-1\}$ there are words $U_r,V_r \in \mathcal W_d$ satisfying
\begin{align*}
w_r = U_rV_r
\end{align*}
and
\begin{align*}
w_{r+1}=V_rU_r.
\end{align*}
Thus it is enough to prove that
\begin{align*}
\mu_{a_1,\dots,a_d}(UV)=\mu_{a_1,\dots,a_d}(VU)
\end{align*}
for arbitrary words $U,V \in \mathcal W_d$.
[/step]
[step:Evaluate one rotation as two reversed products]
Fix words $U,V \in \mathcal W_d$. Let
\begin{align*}
\operatorname{ev}_a:\mathcal W_d \to \mathcal A
\end{align*}
denote the word-evaluation map determined by $\operatorname{ev}_a(X_i)=a_i$ for each $i \in \{1,\dots,d\}$ and by sending the empty word to the unit $1_{\mathcal A}$. Define
\begin{align*}
A_U := \operatorname{ev}_a(U)
\end{align*}
and
\begin{align*}
A_V := \operatorname{ev}_a(V).
\end{align*}
Since evaluation respects concatenation of words, we have
\begin{align*}
\operatorname{ev}_a(UV)=A_UA_V
\end{align*}
and
\begin{align*}
\operatorname{ev}_a(VU)=A_VA_U.
\end{align*}
[guided]
We isolate a single rotation. Choose arbitrary words $U,V \in \mathcal W_d$. The point of introducing an evaluation map is to separate the combinatorics of words from the algebraic calculation in $\mathcal A$.
Let
\begin{align*}
\operatorname{ev}_a:\mathcal W_d \to \mathcal A
\end{align*}
be the word-evaluation map defined by replacing each indeterminate $X_i$ with $a_i$ and replacing the empty word with the unit $1_{\mathcal A}$. This map respects word concatenation: evaluating the concatenated word $UV$ gives the product of the two evaluated words. Therefore, if we define
\begin{align*}
A_U := \operatorname{ev}_a(U)
\end{align*}
and
\begin{align*}
A_V := \operatorname{ev}_a(V),
\end{align*}
then
\begin{align*}
\operatorname{ev}_a(UV)=A_UA_V
\end{align*}
and
\begin{align*}
\operatorname{ev}_a(VU)=A_VA_U.
\end{align*}
This also covers the case where $U$ or $V$ is the empty word, because the empty word evaluates to $1_{\mathcal A}$ and multiplication by $1_{\mathcal A}$ leaves the other evaluated word unchanged.
[/guided]
[/step]
[step:Apply traciality to identify the two moments]
By the definition of the joint moment functional,
\begin{align*}
\mu_{a_1,\dots,a_d}(UV)=\varphi(\operatorname{ev}_a(UV)).
\end{align*}
Using the evaluation identities from the previous step,
\begin{align*}
\mu_{a_1,\dots,a_d}(UV)=\varphi(A_UA_V).
\end{align*}
Since $(\mathcal A,\varphi)$ is tracial, $\varphi(xy)=\varphi(yx)$ for all $x,y \in \mathcal A$. Applying this with $x=A_U$ and $y=A_V$ gives
\begin{align*}
\varphi(A_UA_V)=\varphi(A_VA_U).
\end{align*}
Again using the definition of the joint moment functional,
\begin{align*}
\varphi(A_VA_U)=\mu_{a_1,\dots,a_d}(VU).
\end{align*}
Hence
\begin{align*}
\mu_{a_1,\dots,a_d}(UV)=\mu_{a_1,\dots,a_d}(VU).
\end{align*}
[/step]
[step:Iterate the one-rotation equality along the cyclic chain]
Apply the one-rotation equality to each adjacent pair $w_r,w_{r+1}$ in the finite cyclic chain from the first step. For every $r \in \{0,\dots,m-1\}$,
\begin{align*}
\mu_{a_1,\dots,a_d}(w_r)=\mu_{a_1,\dots,a_d}(w_{r+1}).
\end{align*}
Chaining these equalities gives
\begin{align*}
\mu_{a_1,\dots,a_d}(w)=\mu_{a_1,\dots,a_d}(w_0)=\mu_{a_1,\dots,a_d}(w_m)=\mu_{a_1,\dots,a_d}(v).
\end{align*}
This is the desired cyclic invariance of tracial moments.
[/step]
