Equilibrium on the Riemann Sphere II: Convergence Structure and Topological Compactification

This research program proposes a geometric and operator-theoretic framework for understanding the structure underlying the Riemann Hypothesis. Instead of relying solely on traditional complex analytic techniques applied to the Riemann Zeta Function, we reinterpret Dirichlet partial sums as trajectories embedded on the Riemann Sphere through stereographic projection. Within this geometric setting, the prime number system is modeled as a dynamical orbit structure governed by logarithmic inertial dynamics. In the first stage of the framework, a log-inertial dynamical system is introduced in which the effective mass grows with logarithmic time. This produces a natural phase damping mechanism whose asymptotic behavior reproduces the statistical structure predicted by the Prime Number Theorem. Numerical experiments based on partial sums further indicate that the second derivative in the real direction exhibits sharply localized geometric peaks near the nontrivial zeros. The second stage establishes a more rigorous analytical structure through three complementary perspectives. A measure-theoretic construction places a canonical measure generated by prime images on the sphere, producing a compact accumulation structure at the south pole. Functional analytic methods then define an integral operator derived from this prime measure, whose spectral properties display a critical dichotomy depending on the real part of the complex variable. Finally, a dynamical systems formulation introduces a Lyapunov energy function whose evolution shows monotonic decay and global convergence toward a unique attractor. Taken together, these results suggest that the infinite-dimensional orbit system generated by primes admits a stable and information-preserving structure only on the critical line. Although a direct spectral identification between the operator and the nontrivial zeros remains open, the framework provides a geometric and dynamical skeleton consistent with the program envisioned by the Hilbert–Pólya Conjecture.