What question did this study set out to answer?

To provide a geometric proof of the Riemann Hypothesis using a spherical helix model and its relationship with Maxwell's equations.

May 28, 2026Open Access

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

Key Points

To provide a geometric proof of the Riemann Hypothesis using a spherical helix model and its relationship with Maxwell's equations.
Constructed a spherical helix with specific geometric properties.
Verified geometric identities and constructed mappings using Maxwell's equations.
Applied mirror symmetry between branches of the zeta function to derive critical insights.
Establishes that ζ(s) = 0 in the critical strip implies Re(s) = 1/2.
Numerical verification of the functional equation at multiple test points to 10^{-34} precision.
Provides a new geometric interpretation of Riemann's continuation method from 2D to 3D.

Abstract

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 rₙ = n - 1/2 satisfies the algebraic identity (s) = (2ˢ - 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 = 2z, 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) = w². (3) Left-right mirror structure — the right branch Cₙ is the geometric carrier of (s), the left branch Cₙ^- is the geometric carrier of (1-s) ; both branches share the same vertex V = 1/2 and are mirror-symmetric about V, the algebraic face of the functional equation (s) = (1-s). Main theorem: (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 ( (S²) = 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 = 1/2 in the complex plane. The left-branch proof follows by mirror symmetry (s) (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 (s) = (1-s) at multiple test points to 10^-34 precision. The continuation method follows Riemann's 1859 work using Hankel contours to extend 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.

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