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