Smooth random radial profiles for isotropic Kac polynomials

Unrefereed written-proof candidate for iid isotropic complex Kac coefficients with positive regularly varying moduli, 0<alpha<2. The empirical microscopic radial measure converges in law to a random probability measure with an almost-sure smooth density. Fixed-threshold fractions have full support and positive quadratic count variance. Additional profile endpoint theorems give a common uniform fraction as alpha decreases to zero and a deterministic Gaussian profile as alpha increases to two. These are not joint degree/index limits. Internal analytic audits and finite diagnostics do not constitute formal verification, external peer review or exhaustive priority.