The Grothendieck Constant is Less Than $\frac{\pi }{2\log (1+\sqrt{2})}-{10}^{-5}$

We prove that the Grothendieck constant $K_G<\frac{\pi }{2\log (1+\sqrt{2})}-{10}^{-5}$. This improves on the work of Braverman, Makarychev, Makarychev, and Naor (2011), who proved that $K_G<\frac{\pi }{2\log (1+\sqrt{2})}-\epsilon$ for an unspecified $\epsilon >0$.