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).
Lixin Wang (Mon,) studied this question.