Kuperberg's Six-Cylinder Conjecture
How many pairwise non-overlapping infinite circular cylinders of unit radius can simultaneously touch a unit ball? Kuperberg conjectured in 1990 that the maximum is six.
- Result
- Proved
- Status
- Resolved
- AI contribution
- AI co-developed
- Method
- Computation
- Field
- Discrete geometry
- Posed by
- Włodzimierz Kuperberg
- Year posed
- 1990
- Years open
- 36y
- Solved
- 2026-07-27
- Model
- Claude (version not disclosed)
- Vendor
- Anthropic
- Collaborators
- Ivan Matić, Radoš Radoičić
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 20 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
The proof reduces the upper bound to 2,954,984 exact rational-polynomial cases, each an elementary arithmetic check by a deterministic verifier; the reduction and certificates were developed with Claude.
Verification
Computer-assisted proof with a fully reproducible exact certificate, published as an arXiv preprint. Not yet peer-reviewed.
Source
arXiv:2607.24691 - A computer-assisted proof of Kuperberg's six-cylinder conjecture