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Level A · Machine-checkable Geometry P-kissing-number-lower-bounds

Kissing configurations in dimensions 10–31

Find sphere arrangements that improve the best known lower bounds on kissing numbers in selected dimensions.

Status (as of 2026-11-08): Open. 2025–2026 brought new lower bounds in many dimensions (e.g. τ₁₁ ≥ 604, τ₁₂ ≥ 841, τ₁₉ ≥ 12,460, and new values in dimensions 25–31); most are preprints, so independently verified configurations are valuable.
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Cite 
@misc{cairn-kissing-number-lower-bounds,
  title        = {Kissing configurations in dimensions 10–31},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/kissing-number-lower-bounds}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Beat the record — a computer checks it

Hard Beats the record: More unit vectors at pairwise angles of at least 60° than the best known kissing configuration in some dimension from 10 to 31.

Recent records came from modifying lattice and code constructions and from AI-guided search (2025–2026). Exact integer or rational coordinates make the check exact.

  1. Get a search program — your chatbot writes it from the prompt below
  2. Run it on your computer — it checks its own result and writes a file
  3. Upload the file — a computer checks it, usually within an hour; no reviewers

It happens: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), verified by a computer.

Get a search program in ChatGPT ↗ Upload a construction

Other chatbot? · File format: Header `d: <dimension>`, then one vector per line: d numbers (integers, fractions a/b or decimals) separated by spaces.

Or give your chatbot a small task on this problem instead
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ChatGPT opens in a new tab and starts on the task right away; Claude opens with it filled in.

Current state

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

The problem

The kissing number τ_d is the maximum number of non-overlapping unit spheres in R^d that can touch a central unit sphere. It is known exactly only in a few dimensions (1–4, 8, 24). Elsewhere we know lower bounds from explicit configurations and upper bounds from semidefinite programming.

Recent lower bounds (check Cohn's table before working on a dimension; values checked against Cohn's table on 8 November 2026).

  • Dimension 10: 510; dimension 14: 1932 (Ganzhinov, 2025).
  • Dimension 11: 604 (Pappu, Bianchi, Kwon & Zou, AI agents, arXiv 2606.10402, 2026), after 593 (AlphaEvolve, 2025) and 592 (Ganzhinov).
  • Dimension 12: 841 (Takhanov, Yun & Assylbekov, arXiv 2606.18984, 2026) — the first improvement over 840 (Leech–Sloane, 1971). A fully rigorous check of the minimum distance has been questioned; an exact certificate submitted here would settle it.
  • Dimension 13: 1154 (Zinoviev & Ericson).
  • Dimensions 17, 18, 20, 21: 5730, 7654, 19448, 29768 (Cohn & Li, 2024).
  • Dimension 19: 12,460 (K. Li, "Improved kissing numbers in nineteen and twenty-one dimensions", 2026 preprint), after 11,948 (B. S. Ho, arXiv 2603.10425, 2026, with an exact checker) and 11,692. The same preprint also improves dimension 21.
  • Dimensions 25–31: 197,584 (25) to 251,188 (31) in Cohn's table, from A. Kravatskiy (2026, github.com/alexlegeartis/KissingNumbers), improving "PackingStar" (Ma et al., arXiv 2511.13391, 2025).

Upper bounds have not changed (de Laat & Leijenhorst, 2024). Most 2025–2026 lower bounds are preprints: independently verified configurations are valuable results here.

Submission format: a list of vectors with exact (integer or rational, possibly scaled) coordinates. The checker verifies all vectors have equal norm and all pairwise inner products are at most half the squared norm (angles ≥ 60°). Score = number of vectors for the stated dimension.

History of the bounds

YearResultBy
1971τ₁₂ ≥ 840Leech & Sloane
2025τ₁₁ ≥ 593AlphaEvolve
2026τ₁₉ ≥ 11,948B. S. Ho
2026τ₁₁ ≥ 604Bianchi, Kwon, Pappu, Zou
2026τ₁₂ ≥ 841Takhanov, Yun, Assylbekov
2026τ₁₉ ≥ 12,460K. Li
2026dimensions 25–31: 197,584 to 251,188A. Kravatskiy

Score: number of vectors in a valid kissing configuration of the stated dimension (maximize)· checker kissing