This paper introduces the Topological Pterin-Organized Semiconductor (TOPOS) model, a novel theoretical framework proposing that the origin of life was driven by geometric and topological constraints in Hadean hydrothermal vents. While traditional prebiotic models focus on stochastic chemical assemblies, the TOPOS theory suggests that the interaction between bicyclic pterins and iron-sulfur (Fe-S) clusters within fractal mineral pores created topological singularities. By applying Riemannian geometry and the Poincaré-Hopf Theorem, we demonstrate that a toroidal topology provides a unique mechanism for "interrupted chaos," allowing for the confinement of electronic flows and protection against thermal decoherence. Quantitative analysis using the Berry Curvature and Monte Carlo simulations reveals that such architectures act as "topological insulators" for quantum information. Results indicate that even under conservative activity thresholds (0.0001%), the emergence of protometabolic units becomes a stochastic inevitability over geological timescales. Furthermore, the model proposes that what is defined as biological order emerges from topologically protected quantum coherence, anchored in mineral matrices. This interdisciplinary approach bridges pharmacodynamics, condensed matter physics, and prebiotic chemistry, offering a falsifiable pathway for the transition from mineral geochemistry to autopoietic biological systems.