Theorem \[thm:main\] in Section \[s:main.thm.pf\].

Acknowledgements {#acknowledgements .unnumbered}
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We thank Richárd Rimányi, who introduced us first to the topic of CoHAs and quantum dilogarithm identities in 2014 at Chapel Hill, NC. Further, we acknowledge support in Summer 2019 from an Office of Naval Research Junior NARC grant.

Quivers {#s:quivers}
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Preliminaries {#ss:quivers.prelims}
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A quiver $Q = (Q_0,Q_1,h,t)$ is a directed graph where $Q_0$ is a finite set of vertices, $Q_1$ is a finite set of edges whose elements are called *arrows*, and $h:Q_1 \to Q_0$ and $t:Q_1\to Q_0$ are maps respectively called *head* and *tail*. The maps $h$ and $t$ encode the orientation of the arrows. A *dimension* vector $\gamma = (\gamma(i))_{i\in Q_0}$ is a list of non-negative integers, one for each quiver vertex. In the sequel, we let $D$ denote the monoid of all dimension vectors.[^1] In the rest of the paper, we use the notation $[p]:=\{1,2,\ldots,p\}$ for any $p\in {\mathbb{N}}$, and $e_i \in D$ is the *simple dimension vector* with $1$ at vertex $i$ and zeros elsewhere. Thus, assuming our quiver has $n$ vertices, we identify $Q_0 = [n]$ and write $D {\cong}{\bigoplus}_{i=1}^n {\mathbb{Z}}_{\geq 0}\cdot e_i$. For $\gamma \in D$, we form the *space of quiver 