The Gutenberg-Richter law says that for every magnitude-7 earthquake, there are roughly ten magnitude-6 earthquakes, a hundred magnitude-5s, and so on. The ratio between small and large earthquakes — the b-value — has been treated as an empirical parameter. It's measured from seismic catalogs, and it varies by region: subduction zones have different b-values than transform faults, volcanic areas different from continental interiors. The variation was attributed to stress, heterogeneity, or fluid pressure, but the underlying reason for the power-law distribution itself remained statistically descriptive rather than mechanically explained.
Pan, Zhang, Lund, and Lei derive the b-value from the coupled geometry and mechanics of fault networks. The power-law scaling of rupture area — which follows from the fractal geometry of fault surfaces — combines with the mechanical scaling of slip magnitude to produce the observed frequency-magnitude distribution. The b-value is not a free parameter. It's a consequence of how faults are shaped and how they move.
The derivation reveals that the frequency-magnitude distribution has two distinct branches with a transition governed by fault criticality — how close the fault system is to a state where small ruptures cascade into large ones. Below the critical point, the distribution follows one power law. Above it, another. The kink between them is where the geometric constraint on rupture area interacts with the mechanical constraint on slip.
What was treated as randomness — the stochastic occurrence of earthquakes at different magnitudes — turns out to be structure. The statistical regularity was geometric all along. The faults are not randomly producing earthquakes at random sizes. They are producing earthquakes at sizes determined by their shape.