Katz–Sarnak Equidistribution: Linking L-Function Zero Statistics to Classical Compact Groups — E8 Intelligence Research

FINDING: Katz–Sarnak equidistribution links the statistics of low-lying zeros of L-function families to the eigenvalue statistics of classical compact groups (unitary, symplectic, orthogonal), unifying arithmetic and random matrix theory. | MATH: For a family $\mathcal{F}$ of L-functions with analytic conductors $c(f)$, the 1-level density $W_{\mathcal{F}}(f) = \frac{1}{|\mathcal{F}|}\sum_{f} \sum_{\gamma_f} f\!\left(\frac{\gamma_f \log c(f)}{2\pi}\right) \to \int f(x)\,W_G(x)\,dx$, where $W_G$ is the Fourier transform of the characteristic function of the symmetry group $G \in \{U(N), Sp(2N), SO(\text{even/odd})\}$. Moments: $\frac{1}{|\mathcal{F}|}\sum_f L(1/2,f)^k \sim a_k\, g_k\, (\log c)^ {k(k+1)/2}$ with $g_k$ the Barnes G-function factor $g_k = \prod_{j=0}^{k-1}\frac{j!}{(j+k)!}$ (Keating–Snaith). | CONNECTION: The symmetry classification is a root-system / Weyl-group structure: $U(N)$ (A-type), $Sp(2N)$ (C-type), $SO$ (B/D-type) — the same Cartan classification governing crysta Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com