  In order to probe the magnetic ground state, we have carried out temperature
dependent magnetic Compton scattering experiments on an oriented single crystal
of magnetite (Fe$_3$O$_4$), together with the corresponding first-principles
band theory computations to gain insight into the measurements. An accurate
value of the magnetic moment $\mu_S$ associated with unpaired spins is obtained
directly over the temperature range of 10-300K. $\mu_S$ is found to be
non-integral and to display an anomalous behavior with the direction of the
external magnetic field near the Verwey transition. These results reveal how
the magnetic properties enter the Verwey energy scale via spin-orbit coupling
and the geometrical frustration of the spinel structure, even though the Curie
temperature of magnetite is in excess of 800 K. The anisotropy of the magnetic
Compton profiles increases through the Verwey temperature $T_v$ and indicates
that magnetic electrons in the ground state of magnetite become delocalized on
Fe B-sites above $T_v$.
