Cheeger entropy on finite graphs and exponential selection through one center

We study the selection of first Dirichlet p-eigenfunctions on finite weighted graphs as p ↓ 1. On the cone of nonnegative Cheeger minimizers, a common calibration expresses the Hessian of the first-order entropy as a nonnegative sum over edges. Its full kernel consists of independent scalings on the connected components of the positive support. We deduce a component classification of the entire entropy-minimizing set. The entropy need not have a unique minimizer, even when the interior graph is connected. For a family with arbitrarily many Cheeger leaves joined through one non-Cheeger center, every ordinary power scale remains flat after center elimination. Writing q = p−1, we prove a uniform limit on the closed leaf-mass simplex at the scale qρ^(1/q). The effective functional is a negative weighted geometric mean and determines all selected masses. We also obtain explicit eigenfunction error bounds in terms of the cut gap and weight ratios, and construct any number of nested Cheeger layers. The final selection problem for general graphs with several entropy-minimal components remains open.