Paper N: The Zeta-Vorticity Web and the Geometric Hermite–Biehler Condition

We introduce the Zeta-Vorticity operator Omega(s) := [J, A(s)] associated to the Eisenstein scattering family A(s) = J M_phi on L^2(SL(2,Z)\H), where J is the Krein involution and M_phi is multiplication by the scattering coefficient phi(s) = xi(2s - 1) / xi(2s). We prove that Omega is diagonal in the Eisenstein basis with eigenvalue 2i Im phi(sigma + it), and that the second commutator [J, Omega] vanishes identically — a result proved unconditionally from the functional equation of xi.

The fundamental object is Im phi(sigma + it) as a real harmonic function in the critical strip, whose zero set we call the Vorticity Web W. We prove that under the Riemann Hypothesis, the zero contours of Im phi and Re phi do not intersect in the gap strip sigma in (1/2, 17/30) — the Geometric Hermite-Biehler Condition — and show this provides two new equivalent forms of the Riemann Hypothesis, extending the equivalence chain of the MNZI programme [14] from eight to ten links.

Near each Riemann zero gamma_n, the vorticity web is locally an affine curve through (1/2, gamma_n) tilted at an arithmetic angle psi_n = arg(xi''(2i gamma_n) / xi'(2i gamma_n)), which we compute for the first twenty zeros. The sequence {psi_n} covers the full circle; based on random matrix heuristics and numerical evidence (KS statistic 0.26 against uniform at n = 20), it is conjectured to be equidistributed mod 2pi.

Note on the gap boundary. The upper boundary of the gap strip is taken as sigma = 17/30 throughout this paper, reflecting the current best unconditional bound due to Guth-Maynard [22] (2024), who improved the zero-density exponent to A = 30/13, advancing the zero-free region from the prior Heath-Brown [21] boundary of sigma > 7/12 to sigma > 17/30. All theorems hold for any sigma_0 in (1/2, 1) used as the gap upper boundary; we use 17/30 as the current sharpest unconditional value.