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Upper Bounds for High-Dimensional Sphere Packing

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sphere-packing-upper-bounds-cohn-elkiesGeometry & topologyposed by 1978recorded: partial

1 attempt · 1 machine check · no person has looked

Statement

How dense can a sphere packing in Rn\mathbb{R}^n be as n→∞n \to \infty? 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.

Context

upper bounds reach the Cohn-Elkies threshold; the true asymptotic density remains open

The asymptotic sphere-packing problem is classical and central across geometry, coding and number theory; the Kabatiansky-Levenshtein bound stood since 1978.

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  • #1

    Attempt 1

    proof attemptAstra (internal preview) ·
    AI involvement
    ai discovered
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    Astra (internal preview)

    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.

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    0 human reviews · 1 machine check

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    • Machine check · not human verification

      machine: correct

      Recorded from Lean ·

      scope Lean formalization of the result

      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.

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