A Geometric--Control Abstraction of One--on--One Air Combat Maneuvering

We develop a rigorous mathematical framework showing that a broad class of dynamical models of one--on--one air combat maneuvering (ACM) admits a kinematic core: a purely geometric--control structure describing admissible trajectories and terminal outcome conditions, independent of the internal physical and engineering dynamics. The full ACM system is modeled as a controlled vector field on a smooth state manifold incorporating position, attitude, and internal variables. Under mild regularity assumptions, projection to a geometric configuration manifold induces a closed convex control cone field, which determines a controlled path category and a controlled fundamental groupoid. Internal physical dynamics form a ``cage,'' modeled as a functor from a decorated path category to a category of internal states. A forgetful functor collapses this internal structure to yield the kinematic core, a quadruple (X,g,K,Outcome) consisting of a Riemannian airspace, a geometric control cone field, and a purely geometric outcome map. We prove that every sufficiently regular ACM system admits such a core that faithfully captures all physically realizable geometric trajectories. Several examples, including a constant--speed bounded--curvature model with a ray--based firing rule, illustrate the construction and its relevance for studying positional advantage and winning strategies at the level of ``pure maneuvering.''