AI 中文总结
该研究得出格罗滕迪克常数$K_G$的新上下界，确定其十分位数字为7，采用与以往不同的方法，由人类与AI协作得出。
AI 中文摘要
我们确定了格罗滕迪克常数$K_G$的新上下界：$\frac{6\pi}{11} \le K_G \le \frac{\pi}{2\log(1+\sqrt{2})} - 10^{-4}$。方法上，我们的下界方法与此前工作不同，它通过确定渐近最优Krivine方案的局限性，而非给出间隙实例的显式构造；我们的上界则通过提出并分析首个渐近舍入方案构造得到，此前工作仅考虑低维方案。这些边界共同确定了此前未知的$K_G$的十分位数字为7。这些边界是通过人类与我们设计的长时程AI研究系统的长期协作得出的。
英文摘要
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.
