Topological, Lie-Operator, and Jump-Diffusion Unification of Non-Markovian Systems, Multi-Agent Games, and High-Dimensional Combinatorial Dynamics

Abstract

This monograph establishes a unified mathematical framework bridging non-Markovian queueing systems, robust H ∞ control theory, multi-player differential games, differential topology, persistent homology, and stochastic Langevin jump-diffusion operators. By mapping continuous state evolutions over smooth state manifolds M and high-dimensional discrete combinatorial lattices S ⊂ Z d , we demonstrate that non-commutative operational server constraints and topological void structures act as dual representations of a singular underlying geometry. We establish the Generalized Lie-Operator and Persistence Unification Theorem, proving that optimal multi-agent H ∞ sensitivity minimization, hypoelliptic Hamiltonian vector fields on cotangent bundles T * M , persistent 1D strategy orbits, and global queue equilibrium bounds are dictated by the Betti numbers b k (M) and Morse index bounds of the manifold.