Directional Relational Manifolds: A Stratified Variable-Rank Framework for Relational Geometry

This version substantially revises the mathematical foundations ofDirectional Relational Manifolds. The original unrestricted formulation has been replaced by a stratifiedvariable-rank framework based on a fixed-rank ambient vector bundle, apositive-semidefinite relational metric, effective quotient fibers,stratumwise connections, and explicit rank-transition maps. The previous general Hausdorff-continuity assumption and unconditionaltoroidal-convergence claim have been removed. The revised manuscript includesexplicit rank-collapse and curved-stratum examples, existence results formetric-compatible connections, and rigorous reductions to Riemannian andFisher–Rao information geometry. Toroidal attractors and holonomy-induced recurrent phases are reserved for aseparate future work.