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on Spherical Designs (AlphaEvolve problem 32)

The statement

A spherical $t$ -design on the $d$ -dimensional sphere $S^d \subset \R^{d+1}$ is a finite set of points $X \subset S^d$ such that for any polynomial $P$ of degree at most $t$ , the average value of $P$ over $X$ is equal to the average value of $P$ over the entire sphere $S^d$ . For each $t \in \mathbb{N}$ , let $C(d,t)$ be the minimal number of points in a spherical $t$ -design. Establish upper and lower bounds on $C(d,t)$ that are as strong as possible.

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.