A Topological Hydrodynamic Framework for High-Rank Elliptic Curves: Geometric Realization of the Birch and Swinnerton-Dyer Invariants up to Rank 5

Abstract

We introduce a deterministic topological and fluid-mechanical correspondence for rational points on elliptic curves over Q, extending the framework systematically from Rank 1 through Rank 5. By establishing a Bohr-Sommerfeld-type circulation quantization condition (\Gamma_k = n_k \kappa_0), we map linearly independent Mordell-Weil generators to distinct hydrodynamic circulation modes within a nested toroidal manifold. The canonical Neron-Tate height pairing is shown to correspond directly to localized enstrophy and vortex kinetic energy, while the regulator determinant maps to the mutual hydrodynamic helicity invariant of the flow fields. To resolve the geometric saturation of 3D Euclidean space at Rank 4, we formulate a nested dual-core geometry with a 45-degree bilateral symmetric flux split (+z and -z), satisfying the net momentum conservation constraint (\sum F_z = 0). The framework is stress-tested against boundary choking via a Rankine vortex cutoff radius, angular precession drift, and local Tamagawa reduction factors. Finally, we demonstrate that the model naturally reproduces the cyclic magnetic dynamics, differential shear, and polarity reversal of solar dynamo phenomenology.