Pairwise angular-momentum emergence as PDE closure for ∇ × F = 0: Kepler's laws as topological attractors, revealing the mathematical origin of planetary orbits

The two fundamental fields in nature—gravity and electromagnetism—are both conservative fields, unified by a single mathematical definition: ∇ × F = 0 . Solving this defining partial differential equation (PDE) may therefore uncover the universal topological rules that govern pattern formation, ranging from engineering flows to cosmic structures. However, the PDE is underdetermined. Here, we solve it via an intrinsic closure derived from the equation itself. The closed PDE yields four topological attractors—spirals, circles, ellipses, and straight lines. Kepler's laws follow directly from ∇ × F = 0 as inevitable topological attractors rather than empirical rules. The results demonstrate that the topological rules producing spiral streamlines in an acoustic cell also dictate the elliptical orbits of planets, thereby establishing a unified mathematical origin for patterns across 18 orders of magnitude.