The Gutenberg-Richter law says that for every magnitude step, earthquakes become roughly ten times less frequent. The b-value is the exponent — it quantifies the ratio of small quakes to large ones. For a century, seismologists have measured it, mapped its spatial variation, and used it as an empirical proxy for regional stress. Higher b-value means more small quakes relative to large ones; lower means more large ones. The statistic is informative. Its origin was unclear.
Pan, Zhang, Lund, and Lei (arXiv:2603.06892) show that the b-value is not just a statistical summary but a geometric-mechanical consequence of fault network structure. In three-dimensional fault networks with mainshock-aftershock sequences, the b-value emerges from two scaling relationships: the power-law distribution of fault rupture areas and the scaling of slip with rupture size. The geometry sets the template; the mechanics fills it in.
The frequency-magnitude distribution has two branches. The transition between them is governed by fault criticality and fracture energy dissipation, while the transition magnitude reflects how many faults are triggered in the sequence. Both branches are consequences of structure, not fits to data.
The b-value was never just a measurement. It was a projection of the fault network's geometry into the frequency domain — a statistic that recorded architecture without anyone realizing it was architecture being recorded.