An Equivalent Geometric Construction of the ζ Function: From Euler Nested Circles to Sphere Helix — Two Proof Paths: Gaussian Path and Maxwell Path (V12)

This paper constructs a complete geometric equivalent of the Riemann zeta function and gives two independent proof paths for the Riemann Hypothesis. The geometry is not prescribed a priori — it is forced by the algebraic structure of the Euler series ζ (s) = Σ 1/nˢ. Two symmetry requirements (centers at integer positions, all circles sharing a common point) uniquely determine the vertex V = 1/2 and nested circles Cₙ with radii rₙ = n − 1/2. The algebraic identity ζ (s) = (2ˢ − 1) ^−1 Σ 1/rₙˢ encodes ζ as the power sum of these radii. Riemann's complexification of s introduces phase rotation, producing magnetic lines Mₘ orthogonal to the nested circles, also passing through V. The two orthogonal families form a bipolar coordinate system whose unique three-dimensional reconstruction (via stereographic projection) is a sphere helix, with closure condition ωₛ/ωᵣ = 1/2. From the sphere helix, the proof forks into two independent paths. Path A (Gaussian Path): The yz-projection of the sphere helix yields the Gaussian spiral — each magnetic line Mₘ projects to one discrete spiral arm. An exact double sum identity connects the yz-projection to ζ (s) ·β (s), where β (s) = L (s, χ−₄) is the Gaussian integer L-function. The unique common closure point of all three orthogonal projections is V = (1/2, 0, 0). For fixed s, the arc configuration is uniquely determined with no degrees of freedom; global closure requires geometric consistency across all three projections, which occurs only at V, giving σ = 1/2. This path uses no physics equations. Path B (Maxwell Path): The tangent field B = (−y, x, 0) on the sphere satisfies ∇·B = 0 and ∇×B = 2ẑ, with the poles as the only singularities. The hairy ball theorem (χ (S²) = 2) guarantees that closure events can only occur at singularities, which correspond to σ = 1/2. Both paths start from the same sphere helix and arrive independently at σ = 1/2. V12 extends the V11 framework (DOI: 10. 5281/zenodo. 20398329) by discovering the Gaussian spiral as the third face of the sphere helix and establishing the purely geometric Gaussian proof path alongside the existing Maxwell path. Bilingual (Chinese and English PDFs).