  Combining the measured binding energies of four of the most weakly bound
rovibrational levels of the $^{87}$Rb$_2$ molecule with the results of two
other recent high-precision rubidium experiments, we obtain exceptionally
strong constraints on the atomic interaction parameters in a highly model
independent analysis. The comparison of $^{85}$Rb and $^{87}$Rb data, where the
two isotopes are related by a mass scaling procedure, plays a crucial role.
Using the consistent picture of the interactions that thus arises we are led to
predictions for scattering lengths, clock shifts, Feshbach resonance fields and
widths with an unprecedented level of accuracy. To demonstrate this, we predict
two Feshbach resonances in mixed-spin scattering channels at easily accessible
magnetic field strengths, which we expect to play a role in the damping of
coherent spin oscillations.
