What was already known
Lundow’s earlier published upper bound was 0.457547. Csikvári’s matching-polynomial inequality and Friedland–Schneider’s spectral-radius result were existing ingredients.
AI involvement
GPT-5.6 Sol Ultra suggested the diagonal transfer-layer idea and its link to Csikvári’s inequality. Qidong He developed the proof, compressed computation and Java implementation.
The acknowledged model contribution is the retained diagonal-layer/Csikvari connection on which the improved upper bound rests.
Importance
Improves the published upper bound for the three-dimensional dimer constant.
Limitations
This is an upper bound, not the exact constant. The specified exact computation is reported, but the program and full analytic argument have no documented independent replay.
Technical verification
The paper reports arbitrary-precision integer vector updates and exact cross-multiplication for the (5,5), p=2 finite bound, with upward rounding to six decimals. This checks the numeric part on specified inputs; no completed independent check of the full transfer-layer theorem or program replay is documented.
Independent checking
Matrix-free sector iteration with arbitrary-precision integer vector updates and exact ratio comparison, followed by upward decimal rounding
- Checker
- Qidong He's Java implementation
- Scope
- Finite (m,n)=(5,5), p=2 ratio bound yielding ell_3 <= 0.452130 under Corollary 3.4
- Basis
- Reported by the linked checker
Human understanding
He explains how diagonal layers, Csikvari's inequality, symmetry compression and monotone spectral-radius ratios yield the bound. The unexplained preference for near-square section sizes is a side observation, not a gap in the central mechanism.
Correction and publication history
Improves the published upper bound for the three-dimensional dimer constant.
Read this published revision