  Using the spin and flux evolution of the accreting pulsar 4U 1626$-$67 across
the 2008 torque reversal, we determine the fastness parameter dependence of the
dimensionless torque acting on the pulsar. We find that the dimensionless
torque is qualitatively different from the existing models: it is concave-up
across the torque equilibrium whereas the existing torque models predict a
concave-down (convex) relation with the fastness parameter. We show that the
dimensionless torque has a cubic dependence on the fastness parameter near the
torque equilibrium. We also find that the torque can not attain large values
away from the equilibrium, either in the positive or the negative side, but
saturates at limited values. The spin-down torque can attain a 2.5 times larger
magnitude at the saturation limit than the spin-up torque. From the evolution
of the frequency of quasi-periodic oscillations of 4U 1626$-$67 across the
torque reversal of 1990, we determine the critical fastness parameter
corresponding to torque equilibrium to be $\omega_{\rm c} \simeq 0.75$ within
the framework of the beat frequency model and boundary region model for
reasonable values of the model parameters. We find that the disc magnetosphere
interaction becomes unstable when the inner radius approaches the corotation
radius as predicted by some models, though with a longer timescale. We also
find that there is an unstable regime that is triggered when the fastness
parameter is 0.8 times the critical fastness parameter ($\omega=0.6$ for
$\omega_{\rm c} \simeq 0.75$) possibly associated with an instability observed
in numerical simulations.
