Upper Bounds for High-Dimensional Sphere Packing
How dense can a sphere packing in be as ? The Kabatiansky-Levenshtein upper bound stood for almost fifty years; the new proof improves the asymptotic upper bound all the way down to the Cohn-Elkies linear-programming threshold.
- Result
- Proved (upper bounds reach the Cohn-Elkies threshold; the true asymptotic density remains open)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Discrete geometry
- Posed by
- —
- Year posed
- 1978
- Years open
- 48y
- Solved
- 2026-08-01
- Model
- Astra (internal preview)
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Lean-verified
- Publication
- Announced
- Significance
- 50 / 100
- Disclosed cost
- $182
- Wikipedia
- No dedicated article
What the AI did
Generated by an internal version of OpenAI's Astra: per the announcement, the mathematical arguments were produced by the system (roughly 2,000 dollars of compute at Sol API rates across all ten results), humans prepared the manuscripts with the same model, and the model then formalized the argument in Lean. A narrated reasoning walkthrough is published for each result.
Verification
Kernel-checked Lean 4 certificate in OpenAI's public ten-proofs repository (Lean 4.32, mathlib, `lake build All`), with an independent Comparator checking route. Statement fidelity and community review of the day-old company announcement remain pending.
Sources
OpenAI: Ten advances in mathematics and theoretical computer science