We report new precision measurements of the elastic electron-proton scattering cross section for momentum transfer squared (Q$^2$) up to 15.75~gevsq. These data allow for improved extraction of the proton magnetic form factor at high Q$^2$ and nearly double the Q$^2$ range of direct longitudinal/transverse separated cross sections. A comparison of our results to polarization measurements establishes the presence of hard two-photon exchange in the $e$-$p$ elastic scattering cross section at greater than 95% confidence level for Q$^2$ up to 8 (GeV/c)$^2$.
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