The period-two formula for the affine-quiver quantity $\tilde{K}_n$

Let $\tilde{K}_n$ be the determinant introduced above, namely

$$\tilde{K}_n:= \begin{vmatrix} a_n & h_{n+3} \\ a_{n+4} & h_{n+7} \end{vmatrix}.$$

Here the variables $a_n$ and $h_n$ are the cluster variables associated with the extending vertex and its neighboring data, respectively. Period-two formula for $\tilde{K}_n$. The period $2$ quantity $\tilde{K}_n$ can be expressed as

$$\tilde{K}_n=\frac{a_{n+12}+a_n}{h_{n+6}}.$$

The preceding proposition establishes that $\tilde{K}_n$ is period $2$, with $\sigma(\tilde{K}_n)=\tilde{K}_{n+1}$; the displayed formula is presented as an alternate expression and is not proved in the source.