Equilibrium on the Riemann Sphere III: Closed Manifold Rigidity and Multi-Perspective Equivalence

This paper is the third in a series developing a geometric and operator-theoretic framework for the Riemann Hypothesis and related Millennium Problems. The first paper established the geometric foundation: Dirichlet partial sums embedded on S² via stereographic projection, with log-inertial dynamics reproducing the statistical structure of the Prime Number Theorem. The second paper developed the convergence and compactification structure: canonical prime measures on S², spectral dichotomy of the associated integral operator, and Lyapunov stability toward a unique attractor on the critical line. The present work completes the program by demonstrating that convergence and completeness, viewed from multiple independent perspectives, realise a single conservation principle — and that this principle simultaneously governs three Millennium Problems. The central observation is geometric: the sphere S² is a closed manifold. Once the prime bundle system is embedded on S² and the BF bound saturation condition m²₀₃ₒ䃒 = −1/4 is adopted as the structural axiom, the interior and exterior of the system become two faces of a single indivisible object. Navier–Stokes regularity and the Yang–Mills mass gap emerge as the two sides of this object — not separate problems, but the same conservation law expressed in different physical languages. The Riemann Hypothesis then appears not as a statement about zeros of a complex function, but as the intertwining specification: the unique normalisation σ = 1/2 at which the operator T₁/₂ mediates between the two sides without information loss. Five algebraically independent paths (Paths A–E), Prime Bundle Irreducibility, and Covering Map Rigidity all converge to the same conclusion from different directions — algebraic, entropic, geometric, operator-theoretic, and analytic. This convergence is not coincidental. It reflects the fact that the closed manifold admits only one consistent internal standard: the critical line. The unified principle proposed here is: A closed system with two observable faces and a well-defined intertwining structure must be calibrated at σ = 1/2. The three Millennium Problems are three ways of observing the same coin.