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Level B · Reproducible Geometry P-spherical-designs

Spherical Designs

A spherical t-design on the d-dimensional sphere S^d ⊂ R^d+1 is a finite set of points X ⊂ 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.

From the catalogue. Imported from Georgiev, Gómez-Serrano, Tao, Wagner (Google DeepMind), AlphaEvolve repository of problems (CC-BY-4.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite 
@misc{cairn-spherical-designs,
  title        = {Spherical Designs},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/spherical-designs}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The problem

A spherical -design on the -dimensional sphere is a finite set of points such that for any polynomial of degree at most , the average value of over is equal to the average value of over the entire sphere . For each , let be the minimal number of points in a spherical -design. Establish upper and lower bounds on that are as strong as possible.

AlphaEvolve found a construction better than the previous record (at publication).

What counts as progress

  • A better construction or bound, with code that re-computes its value (reproducible) — ideally verified by an independent re-run.
  • A proof that a known construction is optimal, or a better bound on the other side.
  • Literature: earlier or newer records (literature claims).

Source and licence

Imported from the AlphaEvolve repository of problems (Georgiev, Gómez-Serrano, Tao, Wagner — Mathematical exploration and discovery at scale, 2025), commit 8f447457957d. Text under CC BY 4.0, code under Apache 2.0; reformatted for this page.