The inevitability of dissipation centers in converging charge transport

In a spherical medium with converging current, uniform ohmic conduction causes a diverging dissipation singularity at the center, rendering the uniform state physically unsustainable. We demonstrate that for any saturable current-induced conductivity feedback, the uniform state is not a steady state at any radius. The system must spontaneously form a central dissipation center to mitigate this geometric imposition. We derive its exact analytical profile and prove its global stability. Crucially, the center's width obeys a universal scaling law, growing as the one-fourth power of the enhancement-to-relaxation ratio, independent of the material's maximum conductivity contrast. This geometry-driven, bifurcation-free steady state holds for all strictly positive feedbacks, contrasting with material-driven instabilities that require thresholds or bifurcations. Furthermore, when enhancement accumulates over repeated current scouring, a historical regime emerges where the center expands with the square root of the current density and the fourth root of the scouring events. This provides a clear experimental criterion to distinguish geometry-driven from material-driven structures. These results establish the formation of dissipation centers as a fundamental geometric necessity in converging charge transport, offering a physical basis for structural self-organization in dense aggregates.