We investigated whether one or a few coupling functions can represent best the interaction between the solar wind and the magnetosphere over a wide variety of magnetospheric activity. Ten variables which characterize the state of the magnetosphere were studied. Five indices from ground-based magnetometers were selected, namely Dst, Kp, AE, AU, and AL, and five from other sources, namely auroral power (Polar UVI), cusp latitude (sin(Λ_c)), b2i (both DMSP), geosynchronous magnetic inclination angle (GOES), and polar cap size (SuperDARN). These indices were correlated with more than 20 candidate solar wind coupling functions. One function, representing the rate magnetic flux is opened at the magnetopause, correlated best with 9 out of 10 indices of magnetospheric activity. This is dΦ _MP /dt = v ^4/3 B _T ^2/3sin^8/3( _c/2), calculated from (rate IMF field lines approach the magnetopause, ∼v)(% of IMF lines which merge, sin^8/3( _c/2))(interplanetary field magnitude, B _T )(merging line length, ∼(B _MP /B _T )^1/3). The merging line length is based on flux matching between the solar wind and a dipole field and agrees with a superposed IMF on a vacuum dipole. The IMF clock angle dependence matches the merging rate reported (albeit with limited statistics) at high altitude. The nonlinearities of the magnetospheric response to B _T and v are evident when the mean values of indices are plotted, in scatterplots, and in the superior correlations from dΦ _MP /dt. Our results show that a wide variety of magnetospheric phenomena can be predicted with reasonable accuracy (r > 0.80 in several cases) ab initio, that is without the time history of the target index, by a single function, estimating the dayside merging rate. Across all state variables studied (including AL, which is hard to predict, and polar cap size, which is hard to measure), dΦ _MP /dt accounts for about 57.2% of the variance, compared to 50.9% for E _KL and 48.8% for vBs. All data sets included at least thousands of points over many years, up to two solar cycles, with just two parameter fits, and the correlations are thus robust. The sole index which does not correlate best with dΦ _MP /dt is Dst, which correlates best (r = 0.87) with p ^1/2 dΦ _MP /dt. If dΦ _MP /dt were credited with this success, its average score would be even higher.