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Tightness of the Cohn-Elkies Bound in Dimension 36

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cohn-elkies-bound-dimension-36Geometry & topologyposed by Henry Cohn, Noam Elkies, 2003recorded: disproved

1 attempt · no person has looked

Statement

Can a Cohn-Elkies auxiliary function certify the best known sphere packing in dimension 3636 as optimal? No. An explicit dual-feasible point for the Cohn-Elkies linear program, built from weight-1818 modular forms for Γ0(24)\Gamma_0(24), shows the two-point linear programming bound in dimension 3636 exceeds the density of the Kschischang-Pasupathy packing by a factor of at least 32.9132.91.

Context

rules out the two-point LP method in this dimension; the optimal packing in dimension 36 remains unknown

The Cohn-Elkies linear program is the method behind the dimension 8 and 24 solutions, so where it provably cannot work is a live question in the sphere packing programme.

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  • #1

    Attempt 1

    constructionClaude Fable 5, Claude Opus 4.8, Codex (GPT-5.6) with Rifat Jumagulov ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    Claude Fable 5, Claude Opus 4.8, Codex (GPT-5.6)
    people
    Rifat Jumagulov

    The disclosure reports substantial assistance: the Claude models were used for the construction of the certificate itself, for the verification tooling and for drafting, and Codex was used as an independent cross-check of the certificate computations.

    rules out the two-point LP method in this dimension; the optimal packing in dimension 36 remains unknown

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