A new hyperbolicity wedge and a joint semicircle limit for Jensen polynomials of Riemann's $\xi$-function

Let $$\xi \text{\hspace{0.17em}⁣}(\frac{1}{2}+z)=\sum _{n\ge 0}\frac{\gamma (n)}{n!}{z}^{2n},J^{d,n}(X)=\sum _{j=0}^d\left(\genfrac{}{}{0px}{}{d}{j}\right)\gamma (n+j)X^j\mathrm{.}$$ The Riemann hypothesis is equivalent to the hyperbolicity of $J^{d,n}$ for every $d,n\ge 0$. We prove that there is an absolute constant $K>0$ such that $$n^3{\log }^2(n+2)\ge K{d}^5⟹J^{d,n}\text{ is hyperbolic}\mathrm{.}$$ Along every sequence with $n,d\to \mathrm{\infty }$ in this region, the empirical measure of the naturally centered and scaled zeros also converges to Wigner’s semicircle law. This gives a simultaneous degree–derivative version of the global semicircle consequence of the fixed-degree Hermite limit of Griffin, Ono, Rolen, and Zagier.