A Convex and Combinatorial Analysis of Virtual Multi-Vector Synthesis in Finite Vector Systems

This paper presents a mathematical reinterpretation of virtual multi-vector synthesis defined over finite vector sets. Unlike conventional approaches that treat multi-vector synthesis as an algorithmic technique, the proposed framework characterizes it as a structured problem combining convex geometry, combinatorial selection, and probabilistic averaging. First, it is shown that the set of all realizable virtual vectors coincides with the convex hull of a finite vector set, providing a geometric interpretation of the synthesis process. Based on this observation, a subset-based formulation is introduced, in which virtual vectors are constructed as averages over selected subsets. This formulation allows the synthesis problem to be interpreted as a combinatorial selection problem. Under a uniform subset selection model, closed-form expressions for the expectation and variance of the synthesized vectors are derived. In particular, it is demonstrated that the approximation behavior can be interpreted through the variance structure of subset-averaged vectors, and that increasing the subset size leads to a systematic reduction in variance. Furthermore, the trade-off between approximation accuracy and combinatorial complexity is analyzed, and the existence of an optimal subset size is established. The proposed framework provides a theoretical foundation for understanding multi-vector synthesis as a structured mathematical process, and offers a general perspective applicable to a wide class of approximation problems over finite vector sets.