Unified Geometric Traversal Theorem: A General Framework for Light Paths in Rectangles, Cuboids, and Hyperrectangles

This paper introduces the Unified Geometric Traversal Theorem, a general analytical framework for determining the number of reflections experienced by a ray propagating within bounded geometric domains. Beginning with the classical problem of a light ray reflecting inside a two-dimensional rectangle, the theorem is extended to arbitrary angles of incidence through a tangent-based formulation and further generalized to three-dimensional cuboids and \(n\)-dimensional hyperrectangles. The resulting expressions reduce reflection-heavy trajectories to closed-form arithmetic relations governed by geometric dimensions, angular parameters, and highest common factors. Real-world optical effects, including absorption, reflectivity, refractive index variation, and polarization losses, are incorporated to bridge idealized geometry with physical behavior. Beyond classical optics, the theorem admits computational applications in ray traversal and grid-based simulations, as well as theoretical extensions to higher-dimensional local coordinate charts relevant to brane models, compactification schemes, and string-theoretic frameworks. The Unified Geometric Traversal Theorem thus provides a unified, dimension-independent method for predicting traversal complexity and boundary interaction counts across physical, computational, and theoretical systems. We refer to this arithmetic reformulation of billiard dynamics as the Arithmetic Billiard Reflection System (ABRS), in which reflection counts and exit behavior are determined directly by intrinsic number-theoretic properties of the domain.