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Combining the measured binding energies of four of the most weakly bound rovibrational levels of the ^87Rb_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 ^85Rb and ^87Rb data, where the two isotopes are related by a mass scaling procedure, plays a crucial role. Using the consistent picture of the interactions that thusdoi:10.1103/physrevlett.88.093201 pmid:11864003 fatcat:htldv7bnjjcnlk7mji6bd4zaoa