The non-extensivity parameter q in Tsallis statistics has been used to fit earthquake frequency-magnitude distributions for decades. It works well — q values between 1.4 and 1.8 reproduce the power-law tails observed in global seismicity data. But q has always been treated as a free parameter, adjusted to match each dataset. Sotolongo-Costa and collaborators show it is something more.
In the fragment-asperity model, earthquakes result from interactions between fault fragments. The key assumption is the relationship between stress-bearing interaction energy and contact surface area. A linear relationship — energy proportional to surface — is the simplest possibility. The authors prove it is also unique: only linear scaling produces a closed-form expression for total Tsallis entropy as a function of q.
This turns q from a fitting parameter into a diagnostic. The range 1.4 ≤ q ≤ 1.8 corresponds to the region of steepest entropy variation — the system's sensitivity is highest there. That this range coincides with empirical q values from mainshocks worldwide is now not a coincidence of curve fitting but a consequence of the fragment-asperity mechanics being near criticality.
The shift matters. A fitting parameter tells you what the data looks like. A criticality indicator tells you what the system is doing. If q measures proximity to a critical point, then changes in q over time or space would signal changes in the fault system's mechanical state — not just a different statistical description of the same thing.
The uniqueness result is what gives the interpretation force. If multiple functional forms for the energy-surface relationship produced closed-form entropies, q could be an artifact of model choice. Because only one does, the parameter is pinned to the mechanics.