The nature of the magnetic correlations in Fe-based superconductors remains a matter of controversy. To address this issue, we use inelastic neutron scattering to characterize the strength and temperature dependence of low-energy spin fluctuations in FeTe$_{0.35}$Se$_{0.65}$ ($T_c sim 14$ K). Integrating magnetic spectral weight for energies up to 12 meV, we find a substantial moment ($agt 0.26 mu_B/$Fe) that shows little change with temperature, from below T$_c$ to 300 K. Such behavior cannot be explained by the response of conduction electrons alone; states much farther from the Fermi energy must have an instantaneous local spin polarization. It raises interesting questions regarding the formation of the spin gap and resonance peak in the superconducting state.
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