Information loss of a 9-input majority gate
Let me first explain what the entropy loss is for a gate like AND. The input entropy is $2h(1/2) = 2$ (where $h(p) = -p\log_2p-(1-p)\log_2(1-p)$), while the output entropy is $h(1/4)$ (since the probability that the output is $1$ is $1/4$). Therefore the entropy loss is $2 - h(1/4) \approx 1.189$.

In the case of recursive majority, the output is uniform, so the output entropy is $1$. The input entropy, in contrast, is $9$, and so the entropy loss is $9-1=8$.