Inferring Shannon Capacity of pentagon
From the givens you can only deduce that the Shannon capacity is at least $\sqrt{5}$. For example, it could be that $\alpha(C_5^8)^{1/8} &gt; \sqrt{5}$, which doesn't contradict your data.

As an aside, the Shannon capacity is usually denoted by $\Theta$, whereas $\theta$ is the Lovász function, which is an upper bound on the Shannon capacity. From the givens you can only deduce that $\theta(C_5) \geq \sqrt{5}$.