  By analyzing the dissipative dynamics of a tunable gap flux qubit, we extract
both sides of its two-sided environmental flux noise spectral density over a
range of frequencies around $2k_BT/h \approx 1\,\rm{GHz}$, allowing for the
observation of a classical-quantum crossover. Below the crossover point, the
symmetric noise component follows a $1/f$ power law that matches the magnitude
of the $1/f$ noise near $1\,{\rm{Hz}}$. The antisymmetric component displays a
1/T dependence below $100\,\rm{mK}$, providing dynamical evidence for a
paramagnetic environment. Extrapolating the two-sided spectrum predicts the
linewidth and reorganization energy of incoherent resonant tunneling between
flux qubit wells.
