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on Shannon capacity of the 7-cycle ($C_{9}$)

The statement

Let $\mathcal{C}_{7}$ denote the cycle graph on $7$ vertices. We define $C_{9}$ to be the Shannon capacity of ${\mathcal C}_{7}$ :

C_{9} := \Theta({\mathcal C}_{7}),

where for a graph $G$ , the Shannon capacity $\Theta(G)$ is defined by

\Theta(G) := \sup_{n \ge 1} \alpha(G^{\boxtimes n})^{1/n}.

Here $\alpha(H)$ denotes the independence number of a graph $H$ , and $\boxtimes$ is the strong graph product.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.