Equilibrium on the Riemann Sphere: Toward a Geometric Foundation for the Riemann Hypothesis

We develop a geometric and dynamical framework for studying the non-trivial zeros of the Riemann zeta function ζ (s) via stereographic embedding of Dirichlet partial sums into the Riemann sphere S². The central analytic result is a second σ-derivative criterion: at any point where the partial sum Sₙ (s) vanishes, the Z-coordinate satisfies ∂²Zₙ/∂σ²|σ=1/2 = 2|Lₙ|² ≥ 0, identifying zeros as dynamical stable points of the spherical geometry. We present three independent characterisations of σ = 1/2 as a distinguished critical value: (I) an algebraic derivation from the basal prime pair (2, 3) via a log-time energy balance; (II) a kinetic energy identity in a log-inertial Lagrangian system; and (III) a phase-space volume argument showing that V (σ, x) = Σp≤x p^−2σ ~ log log x uniquely at σ = 1/2, with entropic alignment index H (1/2) = 1 being the sole finite positive value. The logarithmic growth rate of V (1/2, x) coincides, up to a constant, with the variance in Selberg's central limit theorem for log ζ (1/2 + it), establishing a precise dictionary between the spherical geometry and the analytic fluctuation structure of ζ (s). Numerical experiments on the first twenty non-trivial zeros confirm ∂²Zₙ/∂σ² > 0 in both raw and Levinson-mollified form. We identify the principal remaining gap — converting the geometric preference for σ = 1/2 into a rigorous exclusion of off-critical zeros — as an open problem, and propose the two-scale incompatibility of conjugate zero pairs under the functional equation as the most promising direction.