# A Complementarity Partition Theorem for Multifold Conic Systems

Consider a homogeneous multifold convex conic system $$Ax = 0, \; x\in K_1\times \cdots \times K_r$$ and its alternative system $$A\transp y \in K_1^*\times \cdots \times K_r^*,$$ where $K_1,\dots, K_r$ are regular closed convex cones. We show that there is canonical partition of the index set $\{1,\dots,r\}$ determined by certain complementarity sets associated to the most interior solutions to the two systems. Our results are inspired by and extend the Goldman-Tucker Theorem for linear programming.