New Lower and Upper Bounds for the Grothendieck Constant

2026-08-11

AI Summary

This work addresses the long-standing problem of determining the exact value of the Grothendieck constant $K_G$, with a focus on identifying its first decimal digit. By exposing inherent limitations of asymptotically optimal Krivine-type schemes, the authors derive a novel lower bound from this perspective for the first time. Complementarily, they introduce the first asymptotic rounding scheme that integrates high-dimensional probability and optimization theory to rigorously establish an improved upper bound. These advances collectively narrow the known range of $K_G$ to $\frac{6\pi}{11} < K_G < \frac{\pi}{2 \log(1+\sqrt{2})} - 10^{-4}$, thereby conclusively establishing that the tenths digit of $K_G$ is 7—a result that transcends prior approaches reliant on explicit constructions or low-dimensional techniques.

Abstract

We establish new bounds on the Grothendieck constant $K_G$: \[ \frac{6π}{11} \le K_G \le \fracπ{2\log(1+\sqrt2)} - 10^{-4}. \] Methodologically, our lower bound approach differs from previous works by establishing limitations on the asymptotically optimal Krivine schemes, rather than giving explicit constructions of gap instances. Our upper bound is obtained by proposing and analyzing the first asymptotic construction of rounding schemes, whereas previous works only consider low-dimensional schemes. Together, these bounds determine the previously unknown tenths digit of $K_G$ to be $7$. The bounds were discovered by a long-running collaborative effort of humans and a long-horizon AI research system that we engineered.

Problem

• Grothendieck constant
• lower bounds
• upper bounds
• asymptotic analysis
• rounding schemes

Innovation

• Grothendieck constant
• asymptotic rounding schemes
• Krivine schemes
• upper and lower bounds
• human-AI collaboration