Finite-free limit laws for hyperbolic Jensen polynomials of Riemann's xi-function

We study the global zero distribution of shifted Jensen polynomials of Riemann's xi-function in a simultaneous degree-shift limit. Conditional on hyperbolicity, zeros divided by their exact arithmetic mean converge to the mean-one Marchenko-Pastur law when d/(n+d) tends to a positive parameter c. After centering and scaling, the c=0 regime converges to Wigner's semicircle law. The proof uses an exact backward-difference identity, fixed-order sectorial saddle asymptotics for the xi-coefficients, and a connected-hypergraph expansion of finite-free cumulants. For every fixed moment order r, the paper gives an error of order O_r(d^(-1)+(n+d)^(-1)+1/log(n+d+2)). The theorem is conditional on hyperbolicity and does not prove the Riemann hypothesis. A rational-arithmetic verification script for the weighted-hypertree coefficients is included.