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Level A · Machine-checkable Hard Geometry P-green-42

Green's Open Problem 42

Can the Cohn-Elkies scheme be used to prove the optimal bound for circle-packings in 2 dimensions?

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite 
@misc{cairn-green-42,
  title        = {Green's Open Problem 42},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-42}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

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Current state

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

The problem

The question

Can the Cohn-Elkies scheme be used to prove the optimal bound for circle-packings in 2 dimensions?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«42». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem green_42 :
    answer(sorry) ↔ CohnElkiesOptimal 2 (Real.sqrt 3 / 6)

What counts as progress

  • A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
  • Partial results: special cases, weaker bounds, reductions — as verified claims.
  • Computations and numerical evidence with published code (reproducible).
  • Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • [Gr24] Green, Ben. "100 open problems." (2024).
  • [CoEl03] Cohn, Henry, and Noam Elkies. "New upper bounds on sphere packings I." Annals of Mathematics (2003): 689-714.
  • [Vi17] Viazovska, Maryna S. "The sphere packing problem in dimension 8." Annals of mathematics (2017): 991-1015.
  • [CKM17] Cohn, H., Kumar, A., Miller, S., Radchenko, D., & Viazovska, M. (2017). The sphere packing problem in dimension 24. Annals of mathematics, 185(3), 1017-1033.
  • [Sa21] Sardari, Naser Talebizadeh. "Higher Fourier interpolation on the plane." arXiv preprint arXiv:2102.08753 (2021).

Source and licence

Imported from Formal Conjectures (Ben Green's 100 open problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.