Tightness of the Cohn-Elkies Bound in Dimension 36
Can a Cohn-Elkies auxiliary function certify the best known sphere packing in dimension as optimal? No. An explicit dual-feasible point for the Cohn-Elkies linear program, built from weight- modular forms for , shows the two-point linear programming bound in dimension exceeds the density of the Kschischang-Pasupathy packing by a factor of at least .
- Result
- Disproved(see note)
- Status
- Resolved
- AI contribution
- AI co-developed
- Method
- Construction
- Field
- Sphere packing
- Posed by
- Henry Cohn, Noam Elkies
- Year posed
- 2003
- Years open
- 23y
- Solved
- 2026-07-13
- Model
- Claude Fable 5, Claude Opus 4.8, Codex (GPT-5.6)
- Vendor
- Anthropic / OpenAI
- Collaborators
- Rifat Jumagulov
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 25 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
rules out the two-point LP method in this dimension; the optimal packing in dimension 36 remains unknown
What the AI did
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.
Verification
The result is an explicit dual-feasible certificate, so it is checkable in principle by evaluating the constructed function; the paper reports an independent cross-check of the computations by a second model. We have not reproduced it. arXiv preprint, not peer-reviewed.
Source
arXiv:2607.11319 - A dual linear programming bound for sphere packing in dimension 36