We show that Grothendieck's real constant $K_G$ can be upper bounded by projecting vectors onto a random plane through the origin and thresholding a degree five Hermite polynomial. This resolves a conjecture of Braverman-Makarychev-Makarychev-Naor from 2011, who required an extra randomization step in their rounding scheme and proved $K_G<\fracπ{2\log(1+\sqrt{2})}-10^{-500}$. As a corollary of our result, we prove the bound $K_G<\fracπ{2\log(1+\sqrt{2})}-10^{-217}$ by thresholding degree three Hermite polynomials in the plane. We finally give a rigorous computer-assisted proof that $K_G<\fracπ{2\log(1+\sqrt{2})}-10^{-5}$ using interval arithmetic and degree three Hermite polynomial thresholding.

翻译：我们证明了格罗滕迪克实常数 $K_G$ 可通过将向量投影到过原点的随机平面上，并对五次埃尔米特多项式进行阈值化处理而获得上界。这解决了 Braverman-Makarychev-Makarychev-Naor 在 2011 年提出的一个猜想，他们的舍入方案需要额外的随机化步骤，并证明了 $K_G<\fracπ{2\log(1+\sqrt{2})}-10^{-500}$。作为我们结果的一个推论，通过在平面上对三次埃尔米特多项式进行阈值化处理，我们证明了上界 $K_G<\fracπ{2\log(1+\sqrt{2})}-10^{-217}$。最终，我们利用区间算术和三次埃尔米特多项式阈值化方法，通过严格的计算机辅助证明，得到 $K_G<\fracπ{2\log(1+\sqrt{2})}-10^{-5}$。