This paper presents the complex dynamics synthesis of the combat dy-namics series called tensor-centric warfare (TCW; for the first three parts of the series, see [1] [2] [3]), which includes tensor generalization of classical Lanchester-type combat equations, entropic Lie-dragging and commutators for modeling warfare uncertainty and symmetry, and various delta-strikes and missiles (both deterministic and random). The present paper gives a unique synthesis of the Red vs. Blue vectorfields into a single complex battle-vectorfield, using dynamics on Kähler manifolds as a rigorous framework for extending the TCW concept. The global Kähler dynamics framework, with its rigorous underpinning called the Kähler-Ricci flow, provides not only a new insight into the “geometry of warfare”, but also into the “physics of warfare”, in terms of Lagrangian and Hamiltonian structures of the battlefields. It also provides a convenient and efficient computational framework for entropic wargaming.
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