A Geometric Proof of the Riemann Hypothesis: Spherical Helix, Maxwell's Equations, and the Left-Right Mirror Structure of ζ Zeros

This paper presents a geometric proof of the Riemann Hypothesis. The central construction is the spherical helix — sphere centered at (1, 0, 0) with radius 1/2, frequency ratio 1/2, closing from vertex V = (1/2, 0, 0) to antipodal vertex V' = (3/2, 0, 0).Three layers of equivalence work together: (1) Geometric identity of nested circles — the radius sequence r_n = n - 1/2 satisfies the algebraic identity ζ(s) = (2^s - 1)⁻¹ ∑ (n - 1/2)⁻ˢ (verified to machine precision). (2) The spherical helix is the geometric model of stationary Maxwell's equations — the spherical tangent vector field B = (-y, x, 0) satisfies ∇ · B = 0 and ∇ × B = 2ẑ, equivalent to the stationary magnetic field of a uniform current along the symmetry axis (Biot-Savart solution); the (x, y) projection gives the electric equipotentials (nested circles), the (x, z) projection gives the magnetic streamlines (magnetic lines), correspondence via the complex potential f(w) = w^2. (3) Left-right mirror structure — the right branch {C_n} is the geometric carrier of ζ(s), the left branch {C_n^-} is the geometric carrier of ζ(1-s); both branches share the same vertex V = 1/2 and are mirror-symmetric about V, the algebraic face of the functional equation ξ(s) = ξ(1-s).Main theorem: ζ(s) = 0 in the critical strip implies Re(s) = 1/2.Proof path: The right-branch proof uses Maxwell's source-free condition plus the Hairy Ball Theorem (χ(S^2) = 2) to force the singularities of B to lie at {V, V'}; head-to-tail closure-to-zero of vector segments can only occur at singularities, corresponding to σ = 1/2 in the complex plane. The left-branch proof follows by mirror symmetry ζ(s) ↔ ζ(1-s) — the proof structure extends isomorphically, only direction reverses. The complete ζ zero structure (non-trivial + trivial + functional equation symmetry) is rigorously characterized by both branches together.Numerical verification: Euler-equivalent formula to machine precision; functional equation ξ(s) = ξ(1-s) at multiple test points to 10⁻³⁴ precision. The continuation method follows Riemann's 1859 work using Hankel contours to extend ζ from Re(s) > 1 to the entire complex plane, adding one more dimension to the geometric space Riemann left behind — carrying the continuation from the 2D complex plane onward to the 3D spherical helix.