Scalar edge-root existence for \(1<\gamma\le3/2\)

Determine whether the scalar self-consistency equation \(I(a)=K_1(a)\) for the row-independent power-law ensemble has a real solution \(a^*>\sigma_1(\gamma)\) for every \(1<\gamma\le3/2\).

Background

The paper proves existence of a solution $a^*>\sigma_1(\gamma)$ only for $\gamma>3/2$, using the divergence of $K_1(a)-I(a)$ as $a\downarrow\sigma_1$.

For $1<\gamma\le3/2$, both relevant quantities remain bounded at the endpoint, so the intermediate-value argument does not establish a root. The general Matrix Dyson Equation theory guarantees an edge under its assumptions, but does not establish that this edge is represented by the particular scalar root equation.