We present new precision measurements of the elastic electron-proton scattering cross section for momentum transfer (Q(2)) up to 15.75 (GeV/c)(2). Combined with existing data, these provide an improved extraction of the proton magnetic form factor at high Q(2) and double the range over which a longitudinal or transverse separation of the cross section can be performed. The difference between our results and polarisation data agrees with that observed at lower Q(2) and attributed to hard two-photon exchange (TPE) effects, extending to 8 (GeV/c)(2) the range of Q(2) for which a discrepancy is established at > 95% confidence. We use the discrepancy to quantify the size of TPE contributions needed to explain the cross section at high Q(2).
