Rigorous edge analysis near the bulk threshold

Establish rigorously the behavior of the self-consistent edge \(\tau_0(\gamma)\) and resolve the edge structure of the row-independent power-law ensemble throughout \(1/2<\gamma\lesssim0.7\), particularly as \(\gamma\to1/2^+\).

Background

The scalar fixed-point computation becomes numerically unreliable for $\gamma\lesssim0.7$, despite high-precision recomputation and finer grids. This region lies above the rigorously established fourth-moment threshold $\gamma=1/2$, where the auxiliary spectral measure develops increasingly heavy tails.

The paper therefore does not provide a rigorous description of the self-consistent edge or its asymptotics in this range.