From Paper Chain to Klein Bottle: A Topological Framework for Complementary Human–AI Intelligence

We present a topological framework that replaces the XOR framing of human-AI interaction (safety or capability, sovereignty or assistance) with AND: an architecture in which human sovereignty and AI capability coexist as structural features of a single non-orientable surface. We prove (Theorem 1) that on any compact non-orientable surface, the constant eigenfunction is excluded from the solution space, thereby eliminating degenerate consensus. Non-orientable topology removes the Fredholm obstruction to equilibrium existence and guarantees structurally nontrivial equilibria under continuous forcing. Numerical simulation validates the theory across 1000 paired runs (p < 10⁻¹⁶⁵) and 144 parameter conditions. Four applications share the mathematical foundation: a Deliberative Assembly for decision intelligence, a market topology engine, a mind-of-the-buyer system, and a three-register architecture (Primary, Supporting, New) for trustworthy human-AI agent collaboration. The framework operationalises Ubuntu ethics and the indaba deliberative tradition through algebraic topology, wave mechanics on non-orientable surfaces, Active Inference, and quadratic voting theory.