Toward a Mathematics of Directional Inaccessibility: Formalizing the Directional Closure Principle — What Existing Mathematics Already Tells Us, and Where the Boundary of the Unknown Begins

This paper develops the mathematical-foundations companion to the Dimensional Vocabulary Problem by translating the Directional Closure Principle (DCP) into the language of existing mathematics. The DCP states that all physically executable directions available to an observer are closed within the dimensional manifold containing that observer. Rather than proposing a completed new mathematical theory, the paper uses established frameworks from differential geometry, topology, and brane-world physics to map the accessible boundary with precision. Differential geometry provides the tangent bundle as the complete space of manifold-internal directions; topology provides path-connectedness and closure as formal expressions of confinement; and brane-world physics supplies the physical assumption that ordinary matter fields and instruments may be confined to a hypersurface. These frameworks are assembled into a formal statement of directional closure: no observer or instrument confined to a brane-like manifold can access an external degree of freedom through trajectories, processes, or force fields defined only within that manifold. The paper then identifies where existing mathematics ends and formulates six open problems, including the Normal Bundle Problem, Gravity Leakage Signature Problem, Internal Representation Problem, Dimensional Extension Problem, Confinement Universality Problem, and New Geometry Problem. The result is a research programme for Geometric Inaccessibility Studies, aimed at defining the boundary between known mathematics and the possible mathematics of manifold-external direction.