Riemann Hypothesis Reduced to the Herglotz Inequality under Topological Vacuum Electrodynamics

This note records a geometric reduction of the Riemann Hypothesis under Topological Vacuum Electrodynamics. The continuous sheet used for the particle spectrum (a stochastic electromagnetic vacuum cut into force-free Beltrami tubes, with the fine-structure mode count $\alpha^{-1}=4\pi^3+\pi^2+\pi$ and the minor-radius cutoff) is identified with the dynamical partition function of the Reeb flow on the adele class space.

Native closed-orbit periods $\ell(\gamma)=\ln p$, half-density weights $p^{-1/2}$, and the Archimedean digamma kernel arise as geometric identities of that flow. Symplectic volume preservation and the functional equation give real-line unitarity of the scattering matrix. Compression of the unitary scaling group recovers the closed critical strip $0\le\operatorname{Re}(\rho)\le1$. The Clay requirement then reduces to the single unsigned condition

$\operatorname{Im}\Bigl(-\frac{\Xi'(z)}{\Xi(z)}\Bigr)>0\quad\text{for all }\operatorname{Im}(z)>0,$

or any of its classically equivalent forms (Hermite–Biehler, Laguerre, canonical Hamiltonian positivity, Weil positivity, or Li’s criterion).

$|E(z)|>|E(\overline{z})|\quad(\operatorname{Im}z>0),$

$L_\Xi(x)=[\Xi'(x)]^2-\Xi(x)\Xi''(x)>0\quad(x\in\mathbb{R}),$

$\det\mathcal{H}(t)\ge0\quad\text{(almost everywhere)},$

$W(f,f^*)\ge0\quad\text{(for all admissible \(f\))},$

$\lambda_n\ge0\quad\text{(for all \(n\ge1\))}.$

The kinematic layers and the strip bound sit on the same sheet already used for mass and the forces. The Millennium problem is one global positivity condition.