Upper bounds on sphere-packing density in high dimension
the exponent in the best proved bound
- steps
- 4
- by AI
- 1
- since
- 1929
The line is the frontier over time, climbing as the bound goes up: higher is better here. Filled dots are steps that moved it; muted dots are results that did not. Orange dots are catalog entries, results with AI in the loop. Hollow dots are candidates under review and never move the line.Grey dots along the bottom edge are results from before the quantity had a number, placed at the worst end because they have no value on this axis. Dots that would overlap are nudged sideways a few pixels. Hover a dot for its value and attribution.
About this frontier
How dense can a packing of equal balls in be as ? Minkowski's lattices give density at least , and no construction does exponentially better; the upper bounds all have the form , and the question is how large a can be proved. Blichfeldt's 1929 bound has , Levenshtein reached in 1975 through spherical codes, and Kabatiansky and Levenshtein's of 1978 then stood for forty-eight years while Cohn and Elkies's linear program, which reproduces the best bounds in dimensions 8 and 24, was known to be at least as strong but not whether it was stronger. In 2026 its exact asymptotic strength was determined: , the first movement of the exponent since 1978. The true exponent lies somewhere in and where is open; this frontier tracks the lower end.
Every step, newest first
| Date | Value | Who | Model | Status | Source |
|---|---|---|---|---|---|
| 1 Aug 2026 | best The exact asymptotic strength of the Cohn-Elkies linear program: LP_d^{1/d} -> sqrt(e)/(2 pi), with a matching lower bound showing no auxiliary function can do better. Ten-proofs paper, Chapter 1, Theorem 1.1; Lean certificate in openai/ten-proofs. The first change to the exponent since 1978. | OpenAI's Astra (internal preview) | Astra (internal preview) | AI step lean-verified | entry |
| 1978 | Problems of Information Transmission 14 (1978). Delsarte's linear program on the sphere, then a geometric passage from spherical codes to packings. Stood for forty-eight years; Cohn-Zhao 2014, Sardari-Zargar 2024 and Zhao 2024 improved only the lower-order factor. | Grigory Kabatiansky and Vladimir I. Levenshtein | – | historical | source ↗ |
| 1975 | Mathematical Notes 18 (1975), read in the abstract: "delta_n <= 2^{-n(0.5237+o(1))}", from an improved bound on spherical codes. The first exponent above one half. | Vladimir I. Levenshtein | – | historical | source ↗ |
| 1958 | The simplex bound, Proc. London Math. Soc. 8 (1958): a sharper lower-order factor and the bound the field cited for two decades, but the same exponent as Blichfeldt. A step that did not move the line. | Claude Ambrose Rogers | – | historical | source ↗ |
| 1929 | Delta_n <= (n/2 + 1) 2^{-n/2}, Mathematische Annalen 101 (1929). The first exponential upper bound, and its exponent of one half stood for forty-six years. | Hans Frederick Blichfeldt | – | historical | source ↗ |
The 2026 exponent, the 1978 exponent and the fact that nothing between them moved the exponent are all stated in the ten-proofs paper, Chapter 1, and were read there. Levenshtein's 0.5237 was read in the journal abstract. Blichfeldt's and Rogers's bounds are classical and their exponent of one half is textbook; Rogers is kept as a row that did not move the line. Sidel'nikov's 1973 bound is not drawn: its exponent is usually quoted as 0.5096 but could not be confirmed at source from here. The 2026 row is published rather than a candidate because the entry is lean-verified.