Spherical Designs with Square Cardinality

Definitions and mathematical statement

The intended sphere is apparently the two-dimensional unit sphere, but its dimension is not stated in the supplied question. Under this reading let \[ S^2=\{x\in{\mathbb{R}}^3:\|x\|_2=1\},\qquad \sigma(S^2)=4\pi, \] where $\sigma$ is surface area measure. A spherical $n$-design is a finite set $X\subset S^2$ of distinct points such that \[ \frac{1}{|X|}\sum_{x\in X}p(x)=\frac{1}{4\pi}\int_{S^2}p(x)\,{\,\mathrm{d}}\sigma(x) \quad\text{for every }p\in{\mathbb{R}}[x_1,x_2,x_3]\text{ of total degree at most }n. \] The source question, on this interpretation, is whether \[ \forall n\in{\mathbb{Z}}_{\geq0}\quad\exists X_n\subset S^2\quad |X_n|=(n+1)^2\quad\text{and}\quad X_n\text{ is a spherical }n\text{-design}. \] Thus equal weights and exactly $(n+1)^2$ distinct nodes are required; neither arbitrary positive cubature weights nor a bound with an unspecified constant addresses this precise target. "Degree $n$" is read in the usual design sense of all degrees at most $n$. The case $n=0$ is harmless if the intended indexing begins at $n=1$.

The supplied literature triage dated 2026-08-21 reports rigorous computational existence verification through degree 100 and numerical evidence to substantially higher degrees, while reporting that no proof for every degree was located. These finite-degree and numerical results do not settle the universal quantifier in the displayed question; their exact ranges and hypotheses require independent checking in the supplied references.

Short English statement

On the intended $S^2$ interpretation, does every degree $n$ admit an equal-weight spherical design with exactly $(n+1)^2$ distinct points? The supplied review reports finite-degree computational verification, not a proof for all degrees.

Sources

• Local UnsolvedMath registry, permanent ID 30000021, catalogue entry, Spherical Designs with Square Cardinality.
• Geomathematik (2005), Oberwolfach Reports, pp. 1371--1448, DOI https://doi.org/10.4171/owr/2004/27.
• Chen--Frommer--Lang, supplied DOI https://doi.org/10.1007/s00211-010-0332-5; no article title or theorem number is included in the record.
• Supplied reference labeled "spherical design verification": https://arxiv.org/abs/1502.03562.
• Supplied dated literature assessment: checked 2026-08-21; open status and verification ranges are reported from that curation.