  We investigate the high-energy magnetic excitation spectrum of the high-$T_c$
cuprate superconductor Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ (Bi-2212) using Cu
$L_3$ edge resonant inelastic x-ray scattering (RIXS). Broad, dispersive
magnetic excitations are observed, with a zone boundary energy of $\sim$300 meV
and a weak dependence on doping. These excitations are strikingly similar to
the bosons proposed to explain the high-energy `kink' observed in
photoemission. A calculation of the spin-response based on the ARPES-derived
electronic structure and YRZ-quasi-particles reproduces the key features of the
observed magnetic dispersions with no adjustable parameters. These results show
that it is possible to reconcile the magnetic and electronic properties of the
cuprates within a unified framework.
