An Equivalent Geometric Construction of the ζ Function: From Euler Nested Circles to Sphere Helix — Three Argument Paths: Gaussian, Maxwell, and Ptolemaic (V13)

Starting from the series structure ζ (s) = Σ1/nˢ and introducing no a priori geometric assumptions, this paper uniquely constructs the three-dimensional geometric equivalent of the ζ function — a sphere helix — and gives three independent argument paths for the Riemann Hypothesis. The common foundation has two steps. First, the exact identity ζ (s) = (2ˢ−1) ⁻¹ Σ1/rₙˢ (rₙ = n−1/2) encodes the Euler series as a family of co-vertex nested circles with unique common point V = (1/2, 0). Second, Riemann's complexification produces an orthogonal second family (magnetic lines) ; the two families form a bipolar coordinate system, and bijectivity of stereographic projection guarantees the unique three-dimensional reconstruction is a sphere helix with closure condition ωₛ/ωᵣ = 1/2 — the same 1/2 as the vertex. From this unique equivalent object, three independent paths reach σ = 1/2, each reading a different symmetry of the same geometry. Path A (Gaussian, dynamical): the yz-projection carries the Gaussian spiral, connected to ζ (s) ·β (s) by an exact double sum identity; the dynamics give exact 180° mirror symmetry; the unique common closure point of the three projections is V = (1/2, 0, 0). Path B (Maxwell, topological): the tangent field B = (−y, x, 0) satisfies ∇·B = 0; the hairy ball theorem confines closure events to the pole singularities. Path C (Ptolemaic, multiplicative, new in V13): chords on the sphere are concyclic, so Ptolemy's relation — the multiplicative law of chords — holds identically; the conjugate chord product n⁻ᔆ·n⁻⁽¹⁻ᔆ⁾ = n⁻¹ is invariant under σ ↔ 1−σ, whose unique fixed point is σ = 1/2. Three paths, one starting point, one conclusion. V13 supersedes V12: it adds the Ptolemaic path, consolidates the equivalence as a theorem (no longer axiomatic), and records the three-dimensional exclusion of x₀ = 0. Bilingual: Chinese and English PDFs included. Supplement S1 (June 13, 2026): Rigidity of the Sphere Helix. Two theorems strengthen uniqueness to rigidity. Theorem A (Common-Vertex Inversion): inversion centered at V maps the two co-vertex circle families to a numerically pinned Cartesian grid; every admissible carrier inverts to a plane, so the entire Dupin cyclide class collapses to the unique sphere — the common-vertex condition is the spherical signature. The inverted grid coordinates are the reciprocals of the odd and even integers, making the parity decomposition of ζ geometrically visible. Theorem B (Trajectory Projection Rigidity): curve-level projection data, order forcing, functional rigidity, and the closure ratio 1/2 admit exactly one trajectory, with orientation fixed by Riemann's phase. The chain (𝒞, 𝓜) ⇒ S² ⇒ Γ contains no residual geometric freedom. Files: zetaV13SEN. pdf, zetaV13SCN. pdf.