We propose a minimalist framework for background-independent quantum theory based on the 0-sphere (S^0), the simplest nontrivial topological space consisting of two disconnected points. Unlike conventional approaches that rely on a fixed spacetime geometry or manifold structure, our model introduces relational dynamics between the two points as the sole source of structure. Proper time, spatial separation, and causality emerge from sequences of internal transformations without reference to an external background. This framework yields physically significant features, including internal spinor oscillations, subluminal Zitterbewegung with a characteristic velocity approximately equal to 0.04 times the speed of light, and effective gauge-like symmetries. These phenomena arise solely from the algebra of binary transitions and demonstrate the model's capacity to simulate spacetime behavior from a pre-geometric seed. We compare this approach with established background-independent theories such as loop quantum gravity and causal set theory, highlighting both compatibility and conceptual novelty. The 0-sphere model offers a potentially unifying substrate from which complex spacetime structures may emerge, with implications for experimental and theoretical developments in quantum gravity.