The Dimer Constant of the Cubic Lattice
The dimer constant of , the exponential growth rate of perfect matchings of the cubic lattice, has no closed form and is pinned only by bounds. The upper bound improves from Lundow's , standing since 2001, to , via diagonal transfer layers and an inequality of Csikvari relating the spectral radius of the transfer matrix to the constant.
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI co-developed
- Method
- Computation
- Field
- Statistical mechanics
- Posed by
- classical lattice statistics
- Year posed
- —
- Years open
- —
- Solved
- 2026-07-30
- Model
- GPT-5.6 Sol Ultra
- Vendor
- OpenAI
- Collaborators
- Qidong He
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 20 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
a record upper bound; the exact constant remains unknown
What the AI did
The acknowledgement attributes the paper's two key ingredients to the model: the diagonal transfer layers, which replace the symmetry argument special to the rectangular torus, and the connection with Csikvari's inequality.
Verification
Single-author arXiv preprint; not yet peer-reviewed.
Source
arXiv:2607.28810 - A new upper bound on the dimer constant of Z^3