An Equivalent Geometric Construction of the ζ Function: From Euler Nested Circles to Sphere Helix — Two Argument Paths (V12)

This paper constructs the unique three-dimensional geometric equivalent of the Riemann ζ function — a sphere helix — starting from the series structure ζ(s) = Σ 1/nˢ with no a priori geometric assumptions. Every step is uniquely forced: the series plus two symmetry requirements determine the vertex V = 1/2 and nested circles; Riemann's complexification produces orthogonal magnetic lines; together they force the sphere helix with closure condition ωs/ωr = 1/2. Two independent argument paths for the Riemann Hypothesis are given. Path A (Gaussian): the yz-projection yields two orthogonal curve families (concentric circles and arctan spiral arms) whose prime intersection subset is the Gaussian spiral encoding ζ·β; dynamical time parameters give exact 180° mirror symmetry; the unique common point V = 1/2 gives σ = 1/2. Path B (Maxwell): the tangent field B = (−y, x, 0) satisfies ∇·B = 0; the hairy ball theorem confines closure to pole singularities at σ = 1/2. Bilingual: Chinese and English PDFs included.