Let $\mathbb{S}^2$ be the unit sphere in $\mathbb{R}^3$. Claim: There exists no finite subset $A$ of $\mathbb{S}^2$ such that for every $x \in A$ there are at least eight elements in $A$ that are orthogonal to $x$. Question: Is this claim correct, and how can it be proved?
(I have tried many different constructions of such an A, both analytically and numerically, and they all failed, so I am confident that the claim of non-existence is correct, but I am not able to prove this. Even a pointer to relevant related literature would be appreciated.)