The Gutenberg-Richter law says that for every magnitude-7 earthquake, there are roughly ten magnitude-6 earthquakes, a hundred magnitude-5s, a thousand magnitude-4s. The relationship is a power law, and the exponent — called the b-value — is typically close to 1. This regularity has been observed for over 80 years, across every seismic region, at every scale from laboratory fracture to continental plate boundaries.
The b-value has traditionally been interpreted as a property of the crust: heterogeneity, stress state, temperature, fluid pressure. Regional b-values are mapped as diagnostics of tectonic conditions. High b-value means the crust is fragmenting in many small events. Low b-value means stress is concentrating toward fewer, larger ruptures. The b-value is treated as an observable that encodes information about the material.
Pan, Zhang, Lund, and Lei (arXiv:2603.06892, March 2026) show that the b-value is not an independent observable. It emerges from the coupling of two scaling relationships that are separately understood: the power-law distribution of fault rupture areas in a network, and the scaling relationship between rupture area and slip magnitude for individual faults. Neither scaling alone produces the Gutenberg-Richter distribution. The b-value arises from their interaction — from how the geometry of the fault network constrains the mechanics of individual ruptures.
The distinction matters because it changes what the b-value can tell you. If it's an independent property of the crust, then variations in b-value map directly to variations in crustal conditions. If it's a derived quantity from two coupled scalings, then the same b-value can arise from different combinations of fault geometry and slip mechanics. The diagnostic becomes ambiguous — not because the measurement is uncertain, but because the observable is composite.
The model also produces a two-branch frequency-magnitude distribution, with a transition controlled by fault criticality and fracture energy. The transition magnitude reflects the finite population of faults that can be triggered during an aftershock sequence. The smooth power law that Gutenberg and Richter observed is, in this framework, an approximation that holds when you average over both branches.
Pan, Zhang, Lund, and Lei, "On the coupled geometrical-mechanical origin of the earthquake b-value in fault networks," arXiv:2603.06892 (March 2026).