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New Bounds Established for Grothendieck Constant K_G, Determining Tenths Digit
Key points
- New lower bound for K_G is 6π/11.
- New upper bound for K_G is π/(2log(1+√2)) - 10^-4.
- The tenths digit of K_G is now known to be 7.
- Bounds were found using human-AI collaboration.
New Mathematical Bounds
New lower and upper bounds for the Grothendieck constant K_G have been established. The lower bound is set at 6π/11, and the upper bound is π/(2log(1+√2)) - 10^-4. These new calculations refine the understanding of this mathematical constant.
Determining the Tenths Digit
The newly established bounds have allowed for the determination of the tenths digit of K_G, which was previously unknown. The research indicates that the tenths digit of K_G is 7.
Methodological Approach
The lower bound was derived by analyzing limitations on asymptotically optimal Krivine schemes, a departure from previous methods that focused on explicit constructions of gap instances. The upper bound was achieved by proposing and analyzing the first asymptotic construction of rounding schemes, contrasting with prior work that considered only low-dimensional schemes.
Human-AI Collaboration
The discovery of these bounds resulted from a long-running collaborative effort. This collaboration involved human researchers working alongside a long-horizon AI research system engineered for this purpose.
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Primary sources
arXiv 2608.11158Reporting from
Researchers have established new lower and upper bounds for the Grothendieck constant K_G, determining its previously unknown tenths digit to be 7. The new bounds were achieved through a collaborative effort involving human researchers and an AI research system.