A Twelve-Ray Fractional-Edge Process in a Convex Unitary Ensemble — UC Berkeley

We construct an explicit globally strongly convex unitary ensemble whose equilibrium density vanishes as x1/3x^{1/3} at its left endpoint and as (1−x)1/2(1-x)^{1/2} at its right endpoint. A degree perturbation n−N=O(N3/8)n-N=O(N^{3/8}) extends the single equilibrium band through the fractional point on the N−3/4N^{-3/4} left scale, while the remote soft edge moves on the N−5/8N^{-5/8} scale. After centering the coupled endpoint flow, a cubic pullback of the Fokas–Its–Kitaev problem yields a determinant-one 2×22\times2 local Riemann–Hilbert problem on a twelve-ray collision contour with three colliding quarter-root vertices. The cubic genealogy selects an exact C3C_3-equivariant class. We prove unique solvability in that class for compact nonnegative deformation parameters by a folded half-positive identity and an entire-function indicator argument, and establish a collision-uniform Beals–Coifman inverse on the invariant trace space. We then embed the model into a shrinking-disk RH/∂ˉ\bar\partial steepest-descent analysis using an explicit phase decomposition, weighted Cauchy moments, and exact circle matching. The rescaled correlation kernel converges locally, including its diagonal away from the STAR vertices. Uniform diagonal bounds and trace tightness yield trace-norm convergence on bounded windows. The limiting positive contraction therefore defines a determinantal point process, and the bounded-window point-process laws, joint count laws, Fredholm generating functions, and gap probabilities converge. At criticality, K0(X,X)=c0X1/3+O(X−1)K_0(X,X)=c_0X^{1/3}+O(X^{-1}), so the expected count is (3c0/4)L4/3+O(log⁡L)(3c_0/4)L^{4/3}+O(\log L); every fixed strict thinning has Fredholm cost Θ(L4/3)\Theta(L^{4/3}).