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Spherical Designs (AlphaEvolve problem 32)

Spherical Designs (AlphaEvolve problem 32). Current research status is unassessed; the repository records AlphaEvolve’s result on this question.

MSC 05B30 · Other designs, configurations

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository at commit 8f44745 (2026-07-11); the repository records AlphaEvolve results, not current research statusSource & evidence ↓
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Problem statement

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

Lean 4

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Background

Repository result category: world record — AlphaEvolve improved on the best known result recorded at the time. See Section 6.17, in "Mathematical Exploration and Discovery at Scale " paper: arXiv:2511.02864. Experiment notebook.

Progress & source assessment

The repository page records no bounds for this problem.

References
  1. [1]Georgiev, Bogdan; Gómez-Serrano, Javier; Tao, Terence; Wagner, Adam Zsolt. Repository of problems accompanying “Mathematical exploration and discovery at scale” (arXiv:2511.02864), github.com/google-deepmind/alphaevolve_repository_of_problems (commit 8f44745, 11 July 2026). — Spherical Designs (AlphaEvolve problem 32) linkApache-2.0 (repository license); problem pages reproduced with attribution.

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