Paper O: Geostrophic Rigidity of the Eisenstein Scattering Family: A Taylor–Proudman Theorem for the Riemann Zeros

In classical fluid mechanics, the Taylor–Proudman theorem states that in a rapidly rotating, inviscid fluid, velocity perturbations cannot vary along the axis of rotation, forcing fluid structures into rigid “Taylor columns” aligned with that axis. We establish an exact mathematical analogue for the Eisenstein scattering family A (s) = JMφ on L2 (SL (2, Z) ), where J is the Kre ̆ın involution and φ (s) = ξ (2s − 1) /ξ (2s) is the scattering coefficient. Under this analogy: the rotation axis is the critical line σ = 12 ; the fluid velocity is the spectral data of A (s) ; the Coriolis force is the J -involution; and the vorticity is Im φ (σ + it), the eigenvalue of Ω (s) = J, A (s). We prove the Geostrophic Balance Condition: the commutator J, Ω (s) = 0 holds identically (unconditionally, from the functional equation of ξ). This is the spectral analogue of the Taylor–Proudman rigidity condition. Under the Riemann Hypothesis, the zero set of Im φ — the Vorticity Web — has all connected components anchored on the critical line: the Riemann zeros are “pinned” to the rotation axis by geostrophic balance, exactly as Taylor columns are pinned to the rotation axis in rapidly rotating fluids. The paper provides the physical interpretation of the mathematical results of Paper N 1, directed at the mathematical physics community. Sections 1–? ? establish the fluid-dynamic dictionary; Sections 3–4 state and prove the rigidity theorems; Section 7 presents the numerical vorticity field for the first twenty Riemann zeros. MSC (2020): 11M26, 76U05, 47B50, 47A40, 76E20.