We introduce the Zeta-Vorticity operator Omega (s): = J, A (s) associated to the Eisenstein scattering family A (s) = J Mₚhi on L² (SL (2, Z) ), where J is the Krein involution and Mₚhi 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 gammaₙ, the vorticity web is locally an affine curve through (1/2, gammaₙ) tilted at an arithmetic angle psiₙ = arg (xi'' (2i gammaₙ) / xi' (2i gammaₙ) ), which we compute for the first twenty zeros. The sequence psiₙ 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 sigma₀ in (1/2, 1) used as the gap upper boundary; we use 17/30 as the current sharpest unconditional value.
Paul Buchanan (Sun,) studied this question.