The Duality of Singularities and Non-linear Interactions on the Equator within the Universe OS: A Mathematical Theory of Socio-Biological Dynamics Based on Riemann Sphere Topology and Chaotic Systems

This paper extends the "Universe OS" framework to socio-biological dynamics. By projecting the complex plane onto the Riemann sphere and mapping it to the parameter behavior of non-linear differential/difference equations (such as the tent map), we clarify the mathematical mechanisms behind social instability. We prove that "Greed" (expansion/divergence to infinity) and "Apathy" (contraction/convergence to 0) are dual manifestations of identical dead singularities linked via the inversion map f(z)=1/z. Conversely, the "Equator" (the unit circle where |z|=1), where Euler's identity holds under a pure real phase pi, serves as the unique domain of dynamic equilibrium (Love/Balance). Here, unitary existence is preserved while driving infinite non-linear interactions. Social anxiety is formulated as the systemic friction generated when the feedback network is dragged toward the poles. Consequently, expressive agents acting within this phase space function as the physical tuning mechanism that prevents systemic collapse.