Sufficiently dense Kuramoto networks are globally synchronizing

Consider any network of $n$ identical Kuramoto oscillators in which each oscillator is coupled bidirectionally with unit strength to at least $μ(n−1)$ other oscillators. There is a critical value of the connectivity, $μc$⁠, such that whenever $μ>μc$⁠, the system is guaranteed to converge to the all-in-phase synchronous state for almost all initial conditions, but when $μ<μc$⁠, there are networks with other stable states. The precise value of the critical connectivity remains unknown, but it has been conjectured to be $μc=0.75$⁠. In 2020, Lu and Steinerberger proved that $μc≤0.7889$⁠, and Yoneda, Tatsukawa, and Teramae proved in 2021 that $μc>0.6838$⁠. This paper proves that $μc≤0.75$ and explain why this is the best upper bound that one can obtain by a purely linear stability analysis.