This paper establishes the shared mathematical foundation for three applications built on a common topological framework: a Deliberative Assembly for decision intelligence, a Market Topology Engine for consumer sentiment, and a Mind of the Buyer system for product launch prediction. The mathematics is defined once and inherited by each domain. The framework comprises: a Riemannian identity manifold with cultural coordinate patches, non-orientable surfaces (Möbius strip, Klein bottle) that eliminate artificial dichotomies, wave propagation with forced equilibrium analysis and the Fredholm alternative, Weyl-Tang quadratic voting for consensus formation, Active Inference for belief updating, and geodesic interpolation for few-shot inference from partial data. Numerical validation across three surfaces, two metric conditions, 1000 paired runs, and 144 parameter conditions confirms that the topological effect is universal and requires no parameter tuning.