DRM: Directional Relational Manifolds

We introduce Directional Relational Manifolds (DRM), a geometric frameworkin which dimensionality is not assumed to be globally fixed, but emerges fromthe rank and interaction structure of locally active relational directions. A DRM does not require all local directional sets to be embedded in a singlepre-existing metric space. Instead, each state possesses its own localrelational vector space, relational metric, anchor into the state space, andtransport law along admissible trajectories. Continuity is therefore definedinternally through relational transport rather than through an externalHausdorff metric. The relational metric is allowed to be positive semidefinite. Its kernelrepresents collapsed or inactive directions, while its rank defines theeffective local dimensionality. A directional connection is constructed fromrelational transport, and curvature is defined through infinitesimalnon-commutativity or finite-loop holonomy. We further clarify the relation between recurrence and toroidal topology.Boundedness and recurrence alone do not imply that an attractor is a torus.Toroidal closure follows only under additional assumptions, such as theexistence of independent commuting recurrent phase flows with a full-rankperiod lattice. Under regularity, constant-rank, non-degeneracy, and compatibilityconditions, DRM reduces to Riemannian geometry. When the relational metric isthe Fisher Information Matrix, it also reduces to Fisher--Rao informationgeometry. DRM is therefore proposed as a variable-rank relational extensionof classical differential and information geometry.