Optimal Cohn–Elkies packing bounds by dimension
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How close can the Cohn–Elkies two-point linear-programming bound come to the best known sphere-packing center density in dimension 36? The issue is the optimum of the bounding method, which may remain above the density of an actual packing.
Why it matters and what counts as progress
Why it matters: A gap between the LP optimum and a packing density identifies a limit of the two-point bound, not a failure of the packing itself.
What counts as progress: Certify stronger lower bounds on the LP optimum or improve feasible bounds, thereby quantifying what this two-point method can certify.
How the problem developed
- 2003 · Method [1]
Henry Cohn and Noam Elkies Introduced the linear-programming bound for sphere packing and obtained leading upper bounds in dimensions 4 through 36.
- 2020 · Numerical evidence [2]
Nima Afkhami-Jeddi, Henry Cohn, Thomas Hartman, David de Laat and Amirhossein Tajdini Numerical extrapolation suggested that optimizing the Cohn–Elkies program improves the high-dimensional packing exponent, conjecturally to about 0.6044; the authors stated this was not a proof.
- 1 Aug 2026 · Reported claim [3]
OpenAI (Astra) The report proves the exact exponential rate of the optimal Cohn–Elkies LP, giving a sphere-packing upper-bound exponent about 0.6044.
AI contributions 3
A sharp Fourier certificate for planar circle packing
A real radial Schwartz function on the plane has nonnegative Fourier transform, is nonpositive beyond unit distance, and has normalized origin values making the Cohn–Elkies linear-programming bound attain the triangular-packing density π/(2√3).
Ten Advances in Mathematics and Theoretical Computer Science
For the optimal Cohn–Elkies linear-programming packing bound LP_d, OpenAI gives the exponential root limit √(e/(2π)). This yields the sphere-packing upper bound (√(e/(2π))+o(1))ᵈ and corresponding asymptotic Fourier sign-uncertainty radii.
A dual linear programming bound for sphere packing in dimension 36
Jumagulov certifies that the dimension-36 Cohn–Elkies two-point LP optimum is at least 146.1036734821, more than 32.91 times the Kschischang–Pasupathy packing’s center density.
Sources
- Henry Cohn and Noam Elkies, New upper bounds on sphere packings I (2003)
- Nima Afkhami-Jeddi, Henry Cohn, Thomas Hartman, David de Laat and Amirhossein Tajdini, High-dimensional sphere packing and the modular bootstrap (2020)
- OpenAI, Ten advances in mathematics and theoretical computer science
- A sharp Fourier certificate for planar circle packing
- OpenAI planar Fourier-certificate manuscript
- Planar-packing formalization scope
- OpenAI mathematics release account
- Lean proof of the planar Fourier certificate
- Finite Fourier-certificate verifier
- OpenAI, Ten advances: research account
- OpenAI ten-proofs formalization manifest
- Lean proof of the Cohn–Elkies asymptotic
- Sienicki et al., chapter-by-chapter mathematical review
- Formal sphere-packing challenge statements
- Jumagulov, A dual linear programming bound for sphere packing in dimension 36
Website timeline
Publication dates and editorial changes.
OpenAI reports a sharp Fourier certificate for planar circle packing