Cohn–Elkies high-dimensional sphere-packing rate

The problem asks for the exponential decay rate of the Cohn–Elkies linear-programming sphere-packing bound as the dimension tends to infinity. The supplied sources do not identify an original proposer or date.

Known results

• Kabatyanskii–Levenshtein: the earlier bound had exponent $0.59905576\ldots$.
• Numerical extrapolations suggested exponent $0.6044\ldots$, without a proof.
• In dimension $6$, semidefinite programming shows the linear-programming bound is not sharp.

2023 published exact-rate theorem

The Annals paper “New upper bounds on sphere packings I” proves

$$\mathrm{LP}_d=2^{-(\alpha^*+o(1))d},\qquad \alpha^*=\frac12\log_2\!\left(\frac{2\pi}{e}\right)=0.6044\ldots.$$

Thus the requested rate is settled. An OpenAI manuscript states the same theorem, but its independent presentation was reported as awaiting formal peer review.

Current status (as of June 2026): The exact exponential rate is settled at $\alpha^*=\frac12\log_2(2\pi/e)=0.6044\ldots$; no corresponding high-dimensional question remains open.